local time stepping

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Local time stepping applied to implicit-explicit

methods for hyperbolic systems

F. Coquel, Q. L. Nguyen, M. Postel and Q. H. Tran

November 13, 2007

Abstract

In the context of systems of nonlinear conservation laws it can be crucial

to use adaptive grids in order to correctly simulate the singularities of the solu-

tion over long time ranges while keeping the computing time within acceptable

bounds. The adaptive space grid must vary in time according to the local smooth-

ness of the solution. More sophisticated and recent methods also adapt the timestep locally to the space discretization according to the stability condition. We

present here such a method designed for an explicit-implicit Lagrange projec-

tion scheme, addressing physical problems where slow kinematic waves coexists

with fast acoustic ones. Numerical simulations are presented to validate the al-

gorithms in terms of robustness and efficiency.

keywords: Hyperbolic PDE, Timevarying adaptive grid, Local Time Stepping, implicit-

explicit

1 Introduction

Our scope of interest is the accurate numerical simulation of highly nonlinear con-

servative laws. The solutions of such systems of PDEs exhibit localized and moving

singularities which require costly numerical schemes. At the same time realistic ap-

plications often require simulation of the phenomenon on very long time range. As

developed in [1], we are interested in multiphase flows applications presenting the par-

ticularity of two scale wavelengths: a slow kinematic one which is the phenomenon

of interest for the engineers, and fast acoustic waves arising from the highly nonlinear

physics. This will induce very small time steps when treated explicitly. This discrep-

ancy between kinematic and acoustic speeds can be advantageously dealt with in a

Lagrangian projection formulation, as detailed in [10, 9]. This allows to resolve the

acoustic part of the solution with an implicit scheme, therefore relaxing the time step

bound. The kinematic components are solved explicitly, ensuring in turn the requiredprecision.

Laboratoire Jacques-Louis Lions, CNRS et Universite Pierre et Marie Curie, B.C. 187, 75252 Paris

Cedex 05, FranceDepartement Mathematiques Appliquees, Institut Francais du Petrole, 1 et 4 avenue de Bois-Preau,

92852 Rueil-Malmaison Cedex, France

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This setup is a very good candidate for adaptive methods, such as the one intro-

duced by Harten in the 1990s [14], which was specially designed for the speed-up of

finite volume schemes. The gist of the method consists in analyzing the smoothness

of the solution by performing a wavelet-like transformation. In the context of finite

volume schemes this relies on a dyadic hierarchy of discretizations. The size of the

coefficients in the multiscale basis is tested against a threshold parameter. This infor-

mation on the local smoothness of the solution can be used in several manners. It cantrigger different numerical schemes, like limited non linear and costly fluxes in the

vicinity of singularities and linear high order cheaper fluxes in smooth areas. It can

also be used to design an adaptive grid by locally selecting the coarsest level of dis-

cretization beyond which the coefficients are negligible. The adaptive grid must evolve

at each time step in order to follow the displacement of the singularities of the solution.

This evolution strongly relies on the hyperbolic nature of the equations that ensures a

finite speed of propagation. The time step is monitored by the smallest space grid step

and the CFL-like stability condition. Design and study of fully adaptive scheme can be

found in [7, 16]. Use of the multiresolution technique is described at length in [12] and

[1] and is also implemented in the Lagrangian case in [10, 11]. Simulations exhibits

that it can significantly speed-up the computation.

The goal of the present paper is to describe a further enhancement to the mul-

tiresolution technique: the local time stepping. Numerical schemes designed on non

uniform moving or fixed grids can be sped up by using different time steps in

different areas of the grid, according to the local mesh size. First developments in this

direction can be found in [19] for fixed non-uniform grid just before the development

of Adaptive Mesh Refinement technique in [5, 4]. In the context of the multires-

olution technique previously described, it has been developed and implemented for

one-dimensional scalar conservation laws in [17] and extended to the two-dimensional

shallow water equations in [15], with promising CPU time gains. A macro time step,

fit for the coarsest level of discretization, is subdivided into intermediate time steps.

At a given intermediate time step, the solution is updated only on cells belonging to

the current synchronization level or finer. Transition zones are defined in order to dealwith the synchronization of the solution at the border between consecutive levels of

discretization. Straightforward application of this method to our system of PDEs in

the explicit scheme case, was first presented in [11]. It is now extensively tested for

robustness and performances in this paper.

We also endeavor here to adapt the local time stepping idea to our semi-implicit

scheme, where two wave speeds interplay in the sense that fast acoustic waves are

treated implicitly with a CFL condition number much larger than one, and kinematic

waves are treated explicitly, with a time step verifying a standard CFL condition with

a CFL number less than one. The resulting scheme departs from the original one in

[17] and will be described in details. The implicit version of the local time stepping

algorithm presented in [17], or an alternative to it described by the same authors in[18] cannot be readily applied to our problem where we have to deal with transient

fast acoustic waves. In particular, the prediction of the adaptive grid from one time

step to the next needs to take into account the fast wave speed. Another originality of

our setup is the splitting of our operator into a Lagrangian step and an Euler projection

step which requires some extra synchronization during the time integration.

We first present the model equations and the numerical algorithms on a fixed non

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uniform grid. The next section summarizes the multiresolution setup and its appli-

cation to a finite volume scheme, as described at length in [12, 1]. The section 4

is devoted to the local time stepping enhancement in the case of the explicit scheme.

This part makes full use of the notations established in [17]. The semi-implicit scheme,

which requires a new approach, is treated at last in section 5. Numerical simulations

are described in sections 4.3 and 5.2. They allow to compare the efficiency of the differ-

ent schemes with and without local time stepping, in term of accuracy and computingtime.

2 Modeling of the physical problem

Previous works have shown that a relaxation method can be used efficiently for numer-

ical modeling of two-phase flow, see Baudin et al. [2]. Furthermore, it was shown that

a semi-implicit version of the relaxation method is better suited for accurate descrip-

tion of the transport phenomena than the explicit or implicit versions, see Baudin et

al. [3]. A significant speed-up of the computation can be achieved by employing the

adaptive multiresolution (MR) technique, see Coquel et al. [12]. Realistic simulations

in varying geometry pipelines have demonstrated the ability of the model to capturecomplicated physical behavior, including the so-called severe slugging phenomenon

(see Andrianov et al. [1]). We use here a a modified semi-implicit scheme based on a

Lagrangian formulation, for which an explicit stability condition on the time step can

be obtained, ensuring positivity of physical quantities (see [10, 11, 9]).

The density of the mixture , velocity v and the gas mass fraction Y are solution of

the following problem

FindV in V = {V = (,Y, v) R, > 0, Y [0, 1], v R} s.t.

t() + x(u) = 0,

t(Y) + x(Y u) = 0,t(u) + x(u

2 + P) = 0.(2.1)

The thermodynamical closure law P(, Y) appearing in (2.1) can be in real applications

very costly to evaluate, and should satisfy

c2(V) = P(, Y)|Y > 0.

Under this assumption, the system (2.1) is hyperbolic with three distinct eigenvalues

u c < u < u + c. The intermediate eigenvalue corresponds to the slow transport

wave and is linearly degenerated, the remaining ones are much bigger and correspond

to genuinely nonlinear acoustic waves.

Improving the previous numerical treatments of the system (2.1), the scheme weuse here ensures the positivity of the density and physical bounds on the gas mass

fraction. It can be expressed as a flux scheme, which makes local time stepping tech-

niques applicable. The main idea consists in decomposing the flux in an acoustic part,

associated with the genuinely nonlinear waves, and a transport part, associated to the

linearly degenerated waves. This is introduced for instance in [13] using Lagrangian

coordinates for gas dynamics and detailed in [10, 9] in the case of our more involved

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relaxed system of equations. Eventually the Lagrange step where we deal with the

acoustic part of the flux will be treated implicitly, which will enable us to use a larger

time step, basically driven by the transport phenomenon, which will still be treated ex-

plicitly for better accuracy. We will first present the explicit version of the scheme, for

which we adapt the local time stepping enhancement proposed by Muller et al [17].

2.1 Discretization

The Lagrange-Projection splitting is performed at the numerical level in a two-step

scheme which we present in this section without the intermediate details for which we

refer to [10, 9]. We discretize the domain in J cells j = [xj1/2, xj+1/2] of size xj,

such that L =J1

j=0 xj and of center xj = (xj1/2 + xj+1/2)/2 for j = 0, . . . , J 1.

We denote by Vnj

= nj(1, Yn

j, un

j) the numerical solution on cell j at time n and by

Vnj

= nj

(1, Ynj

, unj

) the numerical solution at the end of the Lagrange step.

2.1.1 Explicit scheme

Explicit Euler scheme for the Lagrange step gives

nj

nj

nj

tun

j+1/2un

j1/2

xj= 0,

nj

Ynj

Ynj

t= 0

nj

unj

unj

t+

Pnj+1/2

Pnj1/2

xj= 0,

(2.2)

where denotes the co volume 1/ and unj+1/2

and Pnj+1/2

are the solution of the Rie-

mann problem between states Vnj

and Vnj+1

Pn

j+1/2=

Pnj

+ Pnj+1

2 an

unj+1

unj

2,

unj+1/2

=un

j+ un

j+1

2

Pnj+1

Pnj

2an.

(2.3)

In (2.3), an is a stabilizing coefficient coming from the relaxation formulation of prob-

lem (2.1) as described in [2, 9]. It is set globally for all cells at each time step by the

Whitham condition

a2n > maxj=0,...,J1

P(nj , Y

nj ). (2.4)

The Euler projection step advects the conservative state Vn with the edge velocitiesunj 12

. Combining the two steps together provides the locally conservative flux formu-

lation

Vn+1j = Vnj

t

xj

Fn

j+ 12 Fn

j 12

, (2.5)

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where

Fnj 12

= (0, 0, Pnj 12

)T + (unj 12

)+Vnj1

+ (unj 12

)Vnj

. (2.6)

In the above definition of the fluxes, the superscript (.)+ (respectively (.)) denotes the

positive (respectively negative) part.

We will admit here the following theorem (see [13] and [8])

Theorem 2.1. Suppose that condition (2.4) is satisfied and that

t minj

min(xj, xj1)

max

unj1 annj1 , unj + annj , unj1/2 . (2.7)Then, if n

j> 0 and 0 Yn

j 1 for all j = 0, . . . , J 1, the solution of (2.5) satisfies

n+1j

> 0 and0 Yn+1j

1 for all j = 0, . . . , J 1.

2.1.2 Implicit scheme

The semi-implicit version of the Lagrange-Projection scheme has been derived in [10,

9] and we merely recall the important steps here. Implicitation of the Lagrange step

leads to

nj

nj

nj

tun

j+1/2un

j1/2

xj= 0,

nj

Ynj

Ynj

t= 0,

nj

unj

unj

t+

Pnj+1/2

Pnj1/2

xj= 0,

(2.8)

where the interface quantities arePn

j+1/2=

Pnj

+ Pnj+1

2 an

unj+1

unj

2,

unj+1/2

=u

nj

+ unj+1

2

Pnj+1

Pnj

2an.

(2.9)

We replace Pnj+1/2

andunj+1/2

in (2.8) by their expressions (2.9) and obtain a system for

nj

, unj

and Pnj

. We then eliminate this last unknown, by implicitating the equilibrium

at the end of the Lagrange state Pnj

= P(nj

, Ynj

) and by replacing all (P)j by a global

lower bound an2 defined by (2.4), so that

Pnj

= Pnj an2(n

j

nj). (2.10)

Introducing auxiliary unknowns defined by C,nj

= 12

u

nj

annj

,

C+,nj

= 12

u

nj

+ annj

,

(2.11)

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some straightforward algebra leads eventually to the following system

0 1 + an

nj

annj

C

+,nj1

C+,nj

C+,nj+1

= S +,nj

annj 1 + annj 0 C

,n

j1C

,nj

C,nj+1

= S ,nj(2.12)

where

nj =t

xjnj

(2.13)

and the right hand side is

S+,nj

= C+nj +n

j

2

Pnj + an

2nj Pnj+1 an

2nj+1

,

S ,nj = Cnj

n

j2 Pnj + an2nj Pnj1 an2nj1 . (2.14)

We impose here non reflective boundary conditions, adequate to treat initial value

problems. We will set C,n1

= C,n0

in the first equation for the lower bidiagonal

system, and C+,nJ

= C+,nJ1

in the upper bidiagonal system.

The Euler projection step computes the conservative state at time n + 1:

Vn+1j = Vnj

t

xj

F

n,j+1/2

Fn,j1/2

, (2.15)

where

Fn,j1/2

= 0, 0, Pnj1/2

T+ (u

nj1/2

)+Vnj1

+ (unj1/2

)Vnj

. (2.16)

Introducing the coefficients

mnj =

j

mink=0

wnk,M

n

j =j

maxk=0

wnk,

mnj =J1

mink=j

wnk,M

n

j =J1

maxk=j

wnk

wherew

nj = P

nj + anu

nj ,

wnj = P

nj anu

nj (2.17)

we have the following theorem (see [9])

Theorem 2.2. If nj

> 0 and 0 Ynj

1 for all j = 0, . . . , J 1 and if we impose

condition (2.4), and the following CFL condition on the time step

t =2anx

maxJ1j=0

max

0,M

n

j mnj+1

min

0,mnj

M

n

j+1

(2.18)then the scheme (2.15) has a unique solution which satisfies n+1

j> 0 and0 Yn+1

j 1

for all j = 0, . . . , J 1.

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Additionally to (2.18), we impose to the time step another bound depending solely

on the fast acoustic wave speed, but with a CFL number CFL imp = 10 or even larger,

in order to control the distortion of the acoustic waves.

t CFLimp minj

min(xj1, xj)

max(uj1 aj1, uj + aj). (2.19)

3 Adaptive multiresolution

It is well known that since the fast acoustic waves are treated implicitly, they are

smoothed out very early in the computation (see [12]). The wave of interest which

moves with the slow speed and is computed explicitly, may present on the other hand

singularities that we want to compute as precisely as possible. It is of course natural

to discretize the solution finely in the vicinity of these singularities and more coarsely

elsewhere where it is smooth. In answer to this observation, we have adapted the

multiresolution techniques devised for explicit schemes in [7] and based on ideas in-

troduced in the context of systems of conservation laws by Harten [14] at the beginning

of the nineties. The multiscale analysis of the solution is used to design an adaptive

grid by selecting the correct level out of a hierarchy of nested grids according to thelocal smoothness of the solution. This non-uniform grid evolves with time, with a

strategy based on the prediction of the displacement and formation of the singularities

in the solution. The wavelet basis used to perform the multiscale analysis enables to

reconstruct the solution at any time back to the finest level of discretization, within

an error tolerance controlled by a threshold parameter. The coupling of multiresolu-

tion with the semi-implicit Euler relaxation scheme is detailed in [12] for the non-drift

model and in [1] for the complete model with drift and friction. The adaptation of the

method to the Lagrange projection scheme is straightforward. The important point for

our present work is the enhancement to local time stepping. In the previous works

[7, 12, 1] the time step is dictated by the size of the smallest cell in the adaptive grid

which enters into a CFL stability condition. We first describe how to adapt the localtime stepping approach developed by Muller and Stiriba in [17] in the framework of

the explicit Lagrange projection scheme. The design of the local time stepping algo-

rithm for the implicit version of the Lagrange projection method is the original part of

this work and is described in details.

First we briefly recall in section 3.1 the basics of the multiresolution analysis. It

is then used, in section 3.2, to monitor a time varying adaptive grid for a conservative

finite volume scheme. Technical details and examples can be found in [16].

3.1 Basics of multiresolution analysis

We consider a uniform mesh with step size x, 0j

= [x0j1/2

, x0j+1/2

]. Starting from this

coarsest discretization labeled 0 we define a hierarchy ofK+ 1 nested grids (S k)k=0,...,Kby dyadic refinement, with cells k

j= k+1

2j k+1

2j+1and interfaces xk+1

2j1/2 = xkj1/2

,

with new mid points xk+12j+1/2

= (xkj1/2

+ xkj+1/2

)/2.

Initially, the piecewise constant vector-valued function V is defined on the finest

grid, numbered K, where it is represented by the sequence of its mean values VK =

(VK,j)j on the cells Kj

= [xKj1/2

, xKj+1/2

]. The coarsening operator Pk1k

consists in

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cell averaging from one grid to the coarser one, i.e.,

Vk1 = Pk1kVk with Vk1,j =

1

2(Vk,2j + Vk,2j+1). (3.1)

The inverse operator consists in recovering the mean values on grid level k, given

the mean values on the coarser level k 1. This involves an approximation or

prediction operator Pkk1

which we define here as

Vk = Pkk1Vk1 with

Vk,2j = Vk1,j 18 (Vk1,j+1 Vk1,j1),Vk,2j+1 = Vk1,j + 18 (Vk1,j+1 Vk1,j1). (3.2)We define the detail vector Dk with

Dk,j = Vk,2j Vk,2j. (3.3)The two vectors Vk and (Vk1,Dk) are of same length and we can use Dk along with

Vk1 to recoverVk entirely. Iterating this encoding operation from the finest level down

to the coarsest provides the multiscale representationMK = (V0,D1, . . . ,DK) = MVK.

The indices of the multiscale representation MK vary in

K = {[0, . . . ,N0], [0, . . . ,N0], . . . , [0, . . . , 2K1N0]}

The interest of the multiscale representation lies in the fact that the local regularity of

the function is reflected by the size of its details. We can use this property to compress

the function in the multiscale domain by dropping all details smaller than a given

threshold. To clarify this idea, we first define a threshold operator T acting on the

multiscale representation MK, depending on a subset K of indices = (k, j), by

T(D) =

0 if ,

D otherwise.(3.4)

Given level-dependent threshold values = (k)k=0,...,K, and extending the notationD0 = V0, we introduce the subset = (1, , K) := { s.t. |D| ||}, where

|| = |(k, j)| = k. This completes the definition of the threshold operator T := T ,

and gives rise to an approximating operator A := M1TM acting on the physical

domain representation. In practice, we take advantage of the fact that the remaining

fine-scale details will be concentrated near singularities. This is not such a trivial result

because the operator A is nonlinear since depends on VK through the threshold

scheme. We refer to [6] for a thorough investigation of nonlinear approximation and

the proof of the main result

||VK VK ||L1 < C, with VK = AV

K, (3.5)

valid when k = 2kK. This allows us to define an adaptive grid S where the localsize of the cell will be the grid step corresponding to the finest non negligible detail.

S =(k, j), k {0, . . . , K}, j {0, . . . , 2kN0}, s.t. (k, j/2)

and (k+ 1, j) } , (3.6)

where j denotes the integer part of j.

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The representation V of the solution on this adaptive grid is obtained from the

encoded multiscale representation MK = TMVK by the following

Algorithm 3.1 Partial decoding

Starting from TMV on for level k = 1 K do

if (k, j)

then

compute

Vk,2j using (3.2),Vk,2j = Vk,2j + Dk,j,Vk,2j+1 = 2Vk1,j Vk,2j.

end if

end for

Note, in particular, that the representation by its mean value Vk,j on an intermediate

level kdoes not mean that the function is locally constant on this cell of width 2kx, but

simply that its mean values on the finest grid in this area can be recovered within the

accuracy using the mean values on this intermediate level and the reconstruction

operators Pl

l1for l = k+ 1, . . . , K.

The reverse transformation is performed as follow

Algorithm 3.2 Partial encoding

Starting from V on S for level k = K 1 0 do

if (k+ 1, 2j) then

compute

Vk,j using (3.1),

Dk+1,j = Vk+1,2j Vk+1,2j using (3.2).end if

end for

3.2 Application to a finite volume scheme

We now briefly describe how to use the multiresolution analysis in the context of a

finite volume scheme, written in conservative form on the most refined level of dis-

cretization K as

Vn+1K,j = VnK,j KB

nK,j (3.7)

where K = t/x and the flux balance is defined as

BnK,j = FnK,j+1/2 F

nK,j1/2. (3.8)

The numerical flux FnK,j+1/2 between cells

Kj and

Kj+1 is computed using r values of

the solution on each side of the interface xKj+1/2

FnK,j+1/2 =FVnK,j+1r, . . . ,VnK,j,VnK,j+1, . . . ,VnK,j+r . (3.9)

In our case it will be defined by (2.6) or (2.16). Due to the nestedness of the dyadic

grids, we can define the numerical fluxes on coarser levels kfrom the fluxes defined at

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the same location on finer scales

Fnk,j1/2 = Fnk+1,2j1/2 = . . . = F

nK,2Kkj1/2

. (3.10)

This allows us to define the flux balances by induction for all levels k = 0, . . . , K by

Bnk,j =Fnk,j+1/2 F

nk,j1/2

=Fn

K,2Kk(j+1)1/2 Fn

K,2Kkj1/2

=

2Kk(j+1)1=2Kkj

FnK,+1/2 FnK,1/2

=

2Kk(j+1)1=2Kkj

BnK, =

2Kk1=0

BnK,2Kkj+

. (3.11)

After normalization by the step size 2Kkx at level k, we can apply the same projec-

tion (3.1) and prediction (3.2) schemes on the flux balances as on the solution itself

and we obtain the fully adaptive scheme in 1D as

Vn+1k,j = V

nk,j kB

nk,j, (k, j) S (3.12)

where k = 2kK

K. The important point here is that the adaptive grid S must beadequate to represent the solution at both times n and n + 1. More specifically, if we

denote by n the graded tree obtained by applying A to VnK

, then n can be inflated

into n+1 containing n+1 as well as n, ensuring that estimation (3.5) is valid at bothtimes n and n + 1 when using A = M

1Tn+1 M. Setting n+1 to K does the trickbut it is not very interesting in practice. The inflated tree n+1 should be as small aspossible. An economical way to ensure (3.5) both for Vn and Vn+1 was heuristically

described by Harten in [14]. In [7], we added some gradualness property, necessary to

ensure computational efficiency.

Algorithm 3.3 Prediction of the adaptive grid

n+1

Prediction:for level k = K 1 do

if (k, j) n andDkj k then

(k, j + l) n+1 for l = s, . . . , sifDkj 2k then(k+ 1, 2j) and (k+ 1, 2j + 1) n+1

end if

end if

end for

Gradualness:

for level k = K 1 do

if (k, j) n+1 thenfor || g do

(k 1, j/2 + ) n+1end for

end if

end for

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Let us comment on the choice ofs and g: in Hartens strategy the number of cells

added on each side of important details to account for displacement of the solution

during one time step is s = 1, thanks to the CFL < 1 for explicit schemes. This is

an important point of discussion in the extension of the adaptive algorithm to implicit

schemes with CFL larger than one and the value of the parameter s has been discussed

in [12]. It turns out that s = 1 remains valid in the semi-implicit case becasue the fast

waves are severely damped out by the implicit scheme and can therefore be well rep-resented at the coarser levels of discretization. Concerning the gradualness accounted

for by the second step of algorithm 3.3, the multiresolution stencil should always be

readily available in order to apply the reconstruction formula (3.2) so in our case g = 1.

Below is the actual adaptive algorithm we implement

Algorithm 3.4 Adaptive algorithm

Initialization: encoding of the initial solution and definition of0for time step n = 0, . . . ,N 1 do

Prediction ofn+1 using Algorithm 3.3.Partial decoding ofVn on

S n+1 (derived from

n+1 using (3.6)).

Evolution ofVn to Vn+1 on the adaptive grid S n+1 using (3.12).Definition ofn+1 by partial encoding ofVn+1 .

end for

Decoding ofVN on the finest grid S K.

The evolution step of this algorithm can be implemented in several ways which

have now been extensively studied and compared (see for instance [7] and [1]). We

can now address the purpose of the present paper which is to couple the multiresolution

algorithm with local time stepping.

4 Local time stepping

In the adaptive scheme presented in the previous section the time step is the same

everywhere in the grid and is determined in order to ensure stability. It must obey

some CFL condition and therefore depends on the smallest space grid size x.

We now address the problem of using different time steps depending on the local

size of the adaptive grid cell. We rely on the assumption that if the stability criterion

leads to a time step t on the finest grid of size xK = x at level K, then the scheme

can be applied in coarser regions where the grid size is xk = 2Kkx using a corre-

spondingly larger time step tk = 2Kktand still be stable since the ratio = tk/xk

remains the same (see Figure 1). The obvious difficulty arising from this discrepancy

in time steps is that the solution is not synchronized after one time step and something

special must be done at interfaces between regions of different grid size.We retain here the formalism of [17] and summarize in the first section the algo-

rithm in the explicit case. The second section is devoted to the semi implicit scheme

for which a specific implementation of the local time stepping is designed.

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2

x x

t

t

2

Fig. 1. Using a different time step according to the grid size

4.1 Local time stepping for the explicit scheme

We note t0 = 2Kt the macro time step which can be used on cells at the coarsest

level. The discrete times tn = nt0 are subdivided into 2K intermediate time steps

tn+i2K for i = 1, . . . , 2K with step size t.

At time tn+2Ki the smallest synchronization level is determined by

ki := min{k; 0 k K, i mod 2Kk = 0}.

All cells belonging to levels finer than ki are updated during the intermediate step

i. This partial updating makes uses of sets of indices which will be updated at eachchange of the adaptive grid

The index sets Ck, k = 0, . . . K of cells at level k that can be evolved in time by

one time step with step size tk = 2Kkt, i.e.

for k = K 1 0 do

Ck :=(k, j) S n+1 ;(k+ 1, ) S n+1 : j ,

end for

CK :=(K, j); (K, j) S n+1 .

The range of dependence , determined by the stencil of the 2rpoint flux (3.9)

and the multiscale local prediction (3.2) as

:= {( r)/2 1, . . . , (+ r)/2 + 1} .

The complement sets, or transition zones, i.e.

Ck := Ik\Ck, Ik := (k, j); (k, j) S n+1 .With these definitions, illustrated in figure 2,the evolution of the solution during a

macro time step is performed in a loop on intermediate time steps (see Figure 3). At

each intermediate time step i the solution is updated on all active cells belonging to

levels k ki, using time step tk = t2Kk, except for the cells in the transition

regions where a half time step is used. It is also updated on cells in the transition

region of the immediately coarser level ki 1, using a half time step. (here 2Kki t).

Thanks to the explicit nature of the scheme, and of the flux computation (2.6), we can

use the conservation property depicted in Figure 4 to synchronize the solution in the

manner detailed in the following algorithm

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0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

CK

CK1C

K1

lm m

l

Fig. 2. Definitionof index set CK1 and CK, transition zone CK1 and ranges

of dependence and m for the cells (K, ) and (K, m).

Algorithm 4.1 Synchronized time evolution for time step tn+i2L

for each intermediate time step i = 1, . . . , 2K do

Initialization

for k = K ki do

Update indices sets Ck, Ckend for

Update fluxes on cells that have been modified at previous time step:

for levels k = K ki1 do

for j s.t. (k, j) S n+1 doCompute F

n+(i1)2K

k,j1/2using (2.3-2.6)

end for

end for

for (ki1 1, j) Cki11 s.t. (ki1, 2j 1) S n+1 do

Compute Fn+(i1)2K

ki11,j1/2using (2.3-2.6)

end for

Time evolutionfor levels k = L ki do

use full time step tkfor (k, j) Ck do

Vn+i2K

k,j= V

n+(i1)2K

k,j

F

n+(i1)2K

k,j+1/2 F

n+(i1)2K

k,j1/2

,

end for

end for

for levels k = L ki 1 do

use half time step

for (k, j) Ck do

Vn+i2Kk,j = Vn+(i1)2K

k,j 2 Fn+(i1)2Kk,j+1/2 Fn+(i1)2Kk,j1/2 ,end for

end for

Elsewhere the solution remains unchanged

end for

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4.2 Regriding at intermediate time steps

An important point in the local time stepping algorithm is the evolution of the adaptive

grid. In [17], it is shown that a prediction of the grid from one macro time step to the

next is not efficient when the number of levels is large since the size of the adaptive

grid increases geometrically. The alternative consists in predicting the evolution of the

grid at even intermediate time steps, for all levels above the synchronization level and

perform a partial encoding/decoding of the solution, in the manner described in thefollowing algorithm

Algorithm 4.2 Intermediate regriding

for intermediate time step i = 2, 4, . . . , 2K 2 do

1. Prediction:

for levels k = K ki do

if (k, j) n+i2K

andDkj k then

(k, j + ) n+(i+1)2K for || sifDk

j

2k then

(k+ 1, 2j) and (k+ 1, 2j + 1) n+(i+1)2Kend if

end if

end for

2. Gradualness:


if (k, j) n+(i+1)2K thenif (k 1, j/2 + ) n+i2

K

then

(k 1, j + ) n+(i+1)2K for || gend if

end ifend for

3.

for levels k = ki + 1 K do

Check that we always have g cells at level k 1 between two sets of cells at

level k. Otherwise refine to level k.

end for

end for

for intermediate step i = 2K do

Perform algorithm 3.3 with gradualness g.

Perform step 3.

end for

The prediction s should be equal to the maximum displacement of a singularity in

one time step. In the explicit case, it is therefore equal to 1 as in the standard mul-

tiresolution algorithm 3.3. We will see that for the semi implicit case, it is necessary

to take into account the fast wave and assume a speed larger than 1. The gradualness

parameter g must ensure the possibility of regriding and still preserve the transition

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/2

/2

/2

/2

/2

/2

x

t

i=2 k =1

i=3 k =2

i=4 k =0

i=1 k =21

2

3

4

Fig. 3. A three level adaptive grid with transition zones of width 1 rep-

resented with dotted lines. The intermediate step number is denoted by

i = 1, . . . ,4, ki denotes the synchronization level. The red and green arrowsindicate the fluxes that are computed at the corresponding time step

k,j1

k,j1

k,j1

k+1,2j

k+1,2j

k+1,2j

n

n+1/2

n+1 n+1

n+1/2

n

F

F

Fk,j1/2

n

n+1/2

n

k+1,2j1/2

k+1,2j1/2

k,j2

n

n+1

k,j2

k,j3/2F n

k,j3/2

nF

1

2

k,j3/2

nF1

2

VV

V V

V

V V

V

V

Fig. 4. Flux conservation at interface between level k and k + 1 implies

2Fnk,j1/2

= Fnk+1,2j1/2

+ Fn+1/2

k+1,2j1/2. The solution Vn+

1/2k,j1

is used to compute

Fn+1/2

k+1,2j1/2

zone. It is therefore linked to the parameter s through

g =

s + r

2

+ 1 (4.1)

This rule is illustrated in Figure 5 for speeds 1, 2 and 3. For each example, the pre-

dicted tree is represented at the beginning of the macro time step, with the singlenon negligible detail symbolized with a . Then the resulting adaptive grid S is

represented at the different intermediate time steps, along with the progression of the

singularity, marked with a . The dotted lines indicate the transitions zones.

4.3 Numerical simulations for the explicit scheme

In this section we validate the algorithms on a test case. We compare the uniform finite

volume scheme (U) with two versions of the multiresolution algorithm: (MR) denotes

the algorithm with global time stepping based on the smallest cell, (LTS) denotes the

local time stepping with partial regriding at each intermediate time step.

The simulation is set in a 32km long pipeline. At the initial time the density of the

mixture is 500kg/m3 until x = 16km and 400kg/m3 beyond. The gas mass fraction is

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a)

x

t

S

~

b)

S

Sx

t

~

c)

x

t

S

~

Fig. 5. Grid prediction at intermediate time steps. indicates the tree atthe beginning of the macro time step, with a single non negligible detail.The evolution of the adaptive grid S includes intermediate regriding every

two intermediate time steps. The singularity is marked with a . The grid

propagation speed is (a) s = 1, (b) s = 2, (c) s = 3.

respectively 0.2 and 0.4 and the speed is uniform and equal to 10m/s. The transport

wave moves slowly toward the left at a speed 29m/s while two acoustic waves are

visible on the density and speed components moving in opposite directions at roughly

263m/s and +238m/s.

The pressure the law corresponds to a perfect gas and incompressible liquid

P(,Y) = a2glY

l (1 Y). (4.2)

Figure 6 displays the density field obtained after 42s of propagation with the uniform,

the multiresolution, and the local time stepping schemes. The grid has 5 levels of

refinement, which amounts to J = 4096 grid points on the smallest level of refinement.

We use a constant elementary time step on the finest level throughout the simula-

tion, determined by taking the minimum time step during a previous uniform simula-

tion. For the 4096 points simulations we use t = 0.0096s. The MR and LTS results

are obtained with a threshold parameter = 0.005. The crosses for the multiresolution

and circles for the local time stepping denote the level of refinement used locally to dis-

cretize the solution. It is read on the right hand side vertical axis. In Figure 7 the range

of abscissa is zoomed in in the three regions of interest. The quality of approximation

of the transport wave is the same for the two algorithms. The local time stepping does

a slightly better job than the standard multiresolution as far as the acoustic waves are

concerned. This is because there are less projection steps performed and therefore less

diffusion.

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360

380

400

420

440

460

480

500

520

540

560

0 10000 20000 30000 400000

1

2

3

4

5

6

x

IniUMR

tree

(a)

360

380

400

420

440

460

480

500

520

540

560

0 10000 20000 30000 400000

1

2

3

4

5

6

x

IniU

LTStree

(b)

Fig. 6. Density field and grid refinement at t = 42s, for = 0.005. (a)Explicit multiresolutionscheme (MR) against uniform solution (U) on finest

(5th) level and initial solution (Ini) (b) Local time stepping (LTS) scheme.

The symbols + and indicate the level of refinement used locally, to be read

on the right hand side vertical axis

To study the performance and the robustness of the algorithm we perform a param-eter study. We test two discretizations: J = 4096 points on a hierarchy of 5 levels and

J = 8192 with 6 levels. The range of the threshold goes from 0 to 102. For each

set of parameters we compute the solution at t = 42s with the three algorithms and

calculate the relative error between the adaptive solution reconstructed on the finest

level at the final time and the solution computed on the uniform finest grid. The er-

ror is computed for the component which is sensitive to both transport and acoustic

phenomena and for the gas mass fraction which sees only the transport effect.

E1() =

Jj=0

|j j|

Jj=0

|j|

, E1(Y) =

Jj=0

|Yj Yj|

Jj=0

|Yj|

. (4.3)

We also compute the error with respect to a reference solution computed on a uniform

mesh at the 9th level of discretization (on 65536 points), that is 8 times finer than the

simulation on 8192 points. It will be considered as the exact solution.

Ere f

1() =

Jj=0

|j j|

J

j=0 | j|, E

re f

1(Y) =

Jj=0

|Yj Yj|

J

j=0 |Yj|. (4.4)

In (4.4), j and Yj denote the average of the reference solution on the 8 or 16 cells

covering cell j. In Figure 8 we display the graph of the error (4.3) with respect to

the uniform solution on the finest level versus the threshold parameter. Both schemes

(MR) and (LTS) converge with respect to the threshold parameter which corroborates

the theoretical estimate (3.5).

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360

380

400

420

440

460

480

500

520

540

560

14500 14700 14900 15100 153000

1

2

3

4

5

6

x

IniU

LTSMR

tree LTStree MR

(a)

500

505

510

515

520

525

530

535

540

545

3000 4000 5000 6000 70000

1

2

3

4

5

6

x

IniU

LTSMR

tree LTStree MR

(b)

370

375

380

385

390

395

400

22000 24000 26000 28000 300000

1

2

3

4

5

6

x

IniU

LTSMR

tree LTStree MR

(c)

Fig. 7. Density field and grid refinement at t = 42s. Zoom on (a) transport

wave, (b) left going acoustic wave (c) right going acoustic wave .

1e-14

1e-10

1e-06

0.01

1e-10 1e-06 0.01

E()

MRLTS

(a)

1e-14

1e-10

1e-06

0.01

1e-10 1e-06 0.01

E(Y)

MRLTS

(b)

Fig. 8. Convergence versus for 6 levels. Relative error with respect tothe uniform solution on the 6 th level(4.3) on (a) (b) Y, with the standardmultiresolution (MR) and the local time stepping strategy (LTS)

In Figure 9 we notice that as expected, the error with respect to the reference

solution (4.4) is bounded below by the error between the uniform solution on the 6th

level and the reference solution.

In Figure 10 we study the dependence between the precision and the speed of

the algorithm, by displaying the relative errors between the adaptive and uniform grid

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0.0001

0.001

0.01

0.1

1e-10 1e-06 0.01

Eref()

MRLTS

(a)

0.0001

0.001

0.01

0.1

1e-10 1e-06 0.01

Eref(Y)

MRLTS

(b)

Fig. 9. Convergence versus for 5 levels. Relative error with respect tothe reference solution (4.4) on (a) (b) Y, with the standard multiresolution(MR) and the local time stepping strategy (LTS)

solutions as a function of the gain in computing time. The gain is the ratio between the

computing time required for the uniform scheme on the finest level of discretization

and the computing time required by the multiresolution (MR) or the local time stepping

(LTS) scheme for a given threshold ratio. Each point on these curves corresponds to a

different value of the threshold parameter . The best CPU gain are achieved for the

highest value of, but also correspond to the highest error The CPU gain can be as

high as 10 for a relative error of one per cent for the standard multiresolution and the

local time stepping goes yet more than twice as fast for a given error level.

1e-14

1e-10

1e-06

0.01

0 10 20 30 40

E()

CPU(unif)/CPU(adapt)

MRLTSU

(a)

1e-14

1e-10

1e-06

0.01

0 10 20 30 40

E(Y)


MRLTSU

(b)

Fig. 10. Convergence versus CPU gain for 6 levels. Relative error with

respect to uniform solution on (a) (b) Y.

In Figure 11 we display the error with respect to the reference solution, as a func-

tion of the CPU gain, for both the 4096 and 8192 points simulations. Here again, each

point corresponds to a different value of. The CPU gain is computed with respect to

the CPU for the uniform simulation on the 5th (respectively 6th) level for the multires-

olution (MR) and local time stepping (LTS) simulations on the 5 level-hierarchy (resp.

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6 level-hierarchy). Both (MR) and (LTS) simulations converge to the error level of the

uniform grid simulation when goes to 0 which is roughly 103 for 4096 points and

5.104 for 8192 points. Since the uniform simulation on 8192 points costs roughly 4

times as much as the one on 4096 points, all multiresolution simulations on the 6 level

hierarchy which exhibit a CPU gain superior to 4 and an error below 103 are doing a

better and faster job than the uniform simulation on the 5th level.

0

0.002

0.004

0.006

0.008

0 10 20 30 40

Eref()


MR 5LTS 5MR 6LTS 6

(a)

0

0.002

0.004

0.006

0.008

0 10 20 30 40

Eref(Y)


MR 5LTS 5MR 6LTS 6

(b)

Fig. 11. Convergence versus CPU gain for 5 and 6 levels. Relative error

with respect to reference solution on (a) (b) Y.

5 Local time stepping for the semi-implicit scheme

The main difficulty to implement local time stepping for the semi implicit scheme lies

in the discrepancy between speeds of acoustic and kinematic waves. Actually, if we

use a time step designed on the fast wave speed instead of the slow transport speed as in

(2.18) the Muller and Stiriba algorithm [17] designed for implicit schemes and CFL= 1is convenient. In that case however, we completely miss the advantage of using an

implicit scheme for the Lagrange phase of the algorithm, which is meant to enable us

to use a very large time step. The second stability condition (2.19), which is enforced

with a very large CFL number (CFLimp = 10 to 20), enables us to design the stencil

of influence (S.I.)of a given location with respect to the acoustic waves knowing that

these waves can travel at most CFLimp cells in one time step. At intermediate time

step i all cells in levels k ki are updated and therefore enter the implicit system at the

Lagrangian step. Some additional cells belonging to coarser levels k = ki 1 0 must

be included in the linear systems (2.12) on each side of Di = {Ck}k=ki,...K to compute

the Lagrangian phase (see Figure 12). They are the cells in the stencil of influence

of cells at the border ofDi. Equipped with (2.18) and (2.19) we can safely restrict

this area S .I to a zone of width CFLimp on the borders ofCki not contiguous to Cki ,

measured in number of cells of size xki .

The solution of the Lagrange phase is used to compute the Euler projection fluxes

(2.16) in a manner described in Algorithm 5.4 but only for the cells on which the solu-

tion should actually be updated. As in the explicit scheme these are the cells belonging

to the levels above the current synchronization level ki, plus some contiguous cells in

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the level immediately coarser, forming the transition zone Cki1 (of length r + 1 in

number of cells at level ki 1 in Figure 12). We will see later that cells at the bound-

aries of the transition zone require a special treatment at the Euler projection step. To

locate them more easily, we introduce the distances between a given cell ( k, j) and a

finer level k on its right and on its left

dl(k, j, k) = min

,(k,)Ck j2kk 1

+, (5.1)

dr(k, j, k) = min

,(k,)Ck

(j + 1)2k

k+

. (5.2)

The overall algorithm equivalent of algorithm 4.1 in the semi implicit case is the fol-

lowing

i=1, kt

x

S.I.

C1

1=2

2

i=2, k =1t

x

S.I.

C 0

t

x

S.I

C1 3i=3, k =2

i=4, k =04t

x

Fig. 12. Transition zones (Ck) and implicit stencil S.I.. The intermediate

step number is denoted by i = 1, . . . ,4, the corresponding synchronizationlevel by ki.

Algorithm 5.1 Synchronized time evolution

for intermediate time step i = 1, . . . , 2K do

Lagrange step.

Compute right hand side (2.14) for (2.12) (see Algorithm 5.2).

Compute linear system coefficients for (2.12) and solve linear systems (see Algo-

rithm 5.3).

Compute Euler projection fluxes (see Algorithm 5.4)

Time evolution and synchronization (see Algorithm 5.5)

end for

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The interface quantities are computed at the beginning of each macro time step on

the synchronized solution. The right hand side of (2.12) should be computed according

to the following algorithm

Algorithm 5.2 Right hand side for the linear systems


for (k, j) Ck

do

compute right hand side using (2.14), for a full time step advance tk.

end for

for k < K and (k, j) Ck do

compute right hand side for a half time step advance tk/2.

end for

end for

for levels k = ki 1 0 do

for cells (k, j) s.t. dl(k, j, ki) z or dr(k, j, ki) z do

compute right hand side for a reduced time step advance tki = tk2kki

end for

end for

We denote by S i the stencil of the implicit systems. It can be a union of discon-

nected chunks. The upper and lower bidiagonal matrices M+ and M of the linear

systems are formed as follows

Algorithm 5.3 Linear systems


compute n+i2K

k,jusing tk if (k, j) Ck else using tk/2.

end for

for level k = ki 1 0 do

compute n+i2K

k,jusing tki .

end forImpose non reflective boundary condition at end of blocks :

if (k, j + 1) S i then

M+k;j,j

= (1 + n+i2K

k,j) and M+

k;j,j+1= n+i2

K

k,ielse

M+k;j,j

= 1 and M+k;j,j+1

= 0

end if

if (k, j 1) S i then

Mk;j,j

= (1 + n+i2K

k,j) and M

k;j,j1= n+i2

K

k,jelse

Mk;j,j

= 1 and Mk;j,j1

= 0

end if

In order to compute the Euler projection fluxes (2.16), the interface quantitiesPk,j+1/2 and uk,j+1/2 are computed directly from the Lagrangian phase solution (withthe pressure updated using (2.10), with (2.9).

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Algorithm 5.4 Euler projection fluxes (2.16)


for (k, j) Ck do

Compute Pn+(i1)2K,j1/2

andun+(i1)2K,j1/2

using (2.9).

Save fluxes at the previous intermediate time step for transition zone end points

Compute fluxes using (2.16)

end forend for

for level k = ki 1 do

for (k, j) Ik and dl(k, j, ki) r+ 1 or dr(k, j, ki) r+ 1 doCompute Pn+i2K,

j1/2andun+i2K,

j1/2using (2.9).

Compute fluxes using (2.16)

end for

end for

In the case of the explicit scheme the fluxes at the end of the transition zone Ckare not modified and the conservation property depicted in Figure 4 can be used in a

very simple manner. In case of the semi-implicit scheme, it is not the case anymore,since the solution has been modified during the Lagrangian step, therefore a special

synchronization must be done on levels k, for ki k < K, at the interface between

cells (k, j) and (k, j + 1) ifdl(k, j, k + 1) = r + 1 or at the interface between cells

(k, j 1) and (k, j) ifdr(k, j, k+ 1) = r+ 1.

Ifdl(k, j, k + 1) = r + 1, the solution on cell (k, j) is updated over tk/2 while

the solution on cell (k, j + 1) is updated over tk. The flux Fn+(i1)2K,k,j+1/2

has

been computed using the synchronized Lagrange step solution and can be used

straightforwardly in (2.15) for cell (k, j + 1). However, solution on cell (k, j) has

already been advanced with a half time step at a previous intermediate time step

i = i 2Kk, using flux Fn+(i 1)2K,

k,j+1/2. This quantity has not been tampered with

since then. Therefore, on (k, j) the update formula (2.15) should be replaced by

Vn+i2K

k,j = Vn+(i1)2K

k,i

t

2x

2F

n+(i1)2K,k,j+1/2

Fn+(i 1)2K,k,j+1/2

Fn+(i1)2K,k,j1/2

. (5.3)

Similarly ifdr(k, j, k+1) = r+1, the solution on cell (k, j1) is updated over tk

while the solution on cell (k, j) is updated over tk/2. The flux Fn+(i1)2K,k,j1/2

has

been computed using the synchronized Lagrange step solution and can be used

straightforwardly in (2.15) for cell (k, j 1). However, solution on cell (k, j)

has already been advanced with a half time step at a previous intermediate time

step i = i 2Kk, using flux Fn+(i 1)2K,

k,j1/2. Therefore, on (k, j) (2.15) should be

replaced by

Vn+i2K

k,j = Vn+(i1)2K

k,j

t

2x

F

n+(i1)2K,k,j+1/2

2Fn+(i 1)2K,k,j1/2

+ Fn+(i1)2K,k,j1/2

. (5.4)

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The time evolution algorithm takes into account this special synchronization and also

prepares right hand side corrections for the linear systems at the next intermediate time

step.

Algorithm 5.5 Time evolution and synchronization

for levels k = L ki do

full time step tk

for (k, j) Ck

Vn+i2K

k,j= V

n+(i1)2K

k,j t

x

F

n+(i1)2K,k,j+1/2

Fn+(i1)2K,k,j1/2

end for

for levels k = L ki 1 do

half time step for (k, j) Ck

Vn+i2K

k,j= Vn

+(i1)2K

k,j t

2x

F

n+(i1)2K,k,j+1/2

Fn+(i1)2K,k,j1/2

.

if k >= ki then

special evolution equations should be used at the end of transition zone, (5.4)

ifdr(k, j, k+ 1) = r+ 1 and (5.3) ifdl(k, j, k+ 1) = r+ 1.

end if

Elsewhere the solution remains unchanged

end for

5.1 Grid prediction and regriding

The time step used in the semi implicit scheme is designed according the stability

condition (2.18) which ensures that the slow transport wave can be handled with a

maximum principle by the explicit projection step. For this part of the solution, the

prediction of the tree assuming a displacement of a singularity of at most one cell per

time step is relevant. The fast acoustic waves being treated implicitly at the Lagrange

step, a singularity in this part of the solution will smooth out in all the domain of com-

putation in a single time step, losing of course most of its singular nature. To quantify

this property we study the simple case of the first order forward implicit scheme ap-plied to a linear scalar equation with speed one tU + xU = 0.

Un+1j = Unj (U

n+1j U

n+1j1 ). (5.5)

We consider a solution at time tn built from a single detail at position Jon level K 1

in the multiresolution domain:

V0 = 0, D1 = . . . = DK2 = 0, DK1j = Jj , D

K = 0.

After transformation into the physical space using the decoding operator M1, the

solution is advanced in time using the implicit scheme (5.5) and then transformed back

in the multiscale representation using the encoding operator M. We display in Figure13 (a) different details of this solution on level K 1, as a function of the coefficient

= t/x. The strength of the initial detail DK1J

decreases rapidly and is below 10%

of its initial value for > 10. The next detail in the direction of the propagation DK1J+1

has a significant amplitude for [0, 5]. We then represent the two following details

DK1J1

and DK1J+2

. They are already very small, and all others are negligible. The curve

corresponding to DK1J+1

indicates that detail DK1J

has propagated at a speed higher

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than 1 in a manner that cannot be completely neglected at level K 1. In Figure 13

(b), the details at the coarser level K 2 are displayed. The strength of the detail on

cell J/2 + 1 is comparable to that on the cell J/2, which also means that some of the

solution has actually moved with speed 2. This elementary example illustrates that the

details are damped by the implicit scheme, which is good because it means that they

will be captured by coarser levels in the multiresolution hierarchy. Nevertheless, to

ensure that the regriding at intermediate time step does not deteriorate in the mannerdescribed in Figure 5 (b), we modify the prediction of the grid and assume that the

speed s in the prediction step of Algorithm 4.2 is in fact 3 instead of 1, which will have

the effect of broadening the tree thanks to the gradualness rule (4.1).

0 2 4 6 8 10 12 14 16 18 200.2

0.0

0.2

0.4

0.6

0.8

1.0

J1

J

J+1

J+2

(a)

0 2 4 6 8 10 12 14 16 18 200.10

0.08

0.06

0.04

0.02

0.00

0.02

J/21

J/2

J/2+1

J/2+2

(b)

Fig. 13. Details at time t as a function of (a) dK1J+m and (b) dK2J/2+m for

m = 1, . . . ,2

5.2 Numerical simulations for the implicit scheme

360

380

400

420

440

460

480

500

520

540

560

0 10000 20000 30000 400000

1

2

3

4

5

6

7

x

IniUMR

tree

(a)

360

380

400

420

440

460

480

500

520

540

560

0 10000 20000 30000 400000

1

2

3

4

5

6

7

x

IniU

LTStree

(b)

Fig. 14. Density field and grid refinement computed with the semi-implicitschemes at time t = 41.6s, for = 0.005. (a) multiresolution scheme (b)Local time stepping scheme.

To illustrate the algorithm presented above we use the same test case as in subsec-

tion 4.3 for the explicit scheme. In the case of the semi-implicit scheme the coefficient

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360

380

400

420

440

460

480

500

520

540

560

14500 14700 14900 15100 153000

1

2

3

4

5

6

7

x

IniU

LTSMR

tree LTStree MR

(a)

370

375

380

385

390

395

400

22000 24000 26000 28000 300000

1

2

3

4

5

6

7

x

IniU

LTSMR

tree LTStree MR

(b)

500

505

510

515

520

525

530

535

540

545

4000 5000 60000

1

2

3

4

5

6

7

x

IniU

LTSMR

tree LTStree MR

(c)

Fig. 15. Density field computed with the semi-implicit schemes at time

t = 41.6s, for = 0.01. Zoom on (a) kinematic wave (b) right goingacoustic wave (c) left going acoustic wave.

1e-08

1e-06

1e-04

0.01

1e-10 1e-06

E()

MRLTS

(a)

1e-08

1e-06

1e-04

0.01

1e-10 1e-06

E(Y)

MRLTS

(b)

Fig. 16. Error on (a) density, (b) gas mass fraction with respect to uniform

solution versus threshold parameter for 6 levels grid hierarchy.

an must be set globally on the all x range at each time step. The uniform scheme

provides us with a time step ensuring stability throughout the simulation t = 0.065s

hence roughly seven times larger than the one used for the explicit schemes simula-

tions.

Figure 14 displays the density field computed with the uniform scheme and the

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1e-08

1e-06

1e-04

0.01

0 4 8 12 16 20

E()


MRLTSU

(a)

1e-08

1e-06

1e-04

0.01

0 4 8 12 16 20

E(Y)


MRLTSU

(b)


solution versus CPU gain for 6 levels grid hierarchy.

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 4 8 12 16 20

Eref()


MR 5LTS 5MR 6LTS 6

(a)

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 4 8 12 16 20

Eref(Y)


MR 5LTS 5MR 6LTS 6

(b)


reference solution versus CPU gain for 5 and 6 levels grid hierarchy.

adaptive schemes (MR) and (LTS) The zoom on the three different waves displayed in

Figure 15 show the robustness of the new algorithm. The kinematic wave is reproduced

as well, if not better as with the standard (MR) scheme. The acoustic waves are also

well handled even though they are more smoothed out with the local time stepping

algorithm than with the standard multiresolution.

A parameter study for varying values of the threshold is presented in Figures 16

to 18. In Figure 16 we display the relative error (4.3) between the density or the gas

mass fraction obtained using the uniform scheme or the different adaptive schemes. As

in the explicit case, the MR and LTS schemes both exhibit a monotonous convergence

in O(). We also observe that the relative error on the gas mass fraction is two ordersof magnitude smaller than the error on the density. This is quite normal since the gas

mass fraction is driven by the slow kinetic wave and actually explicitly computed in

the semi-implicit scheme. The adaptive grid is retains almost all details in the vicinity

of the discontinuity in the gas mass fraction, which accounts for a much smaller error.

On the other hand the density sees acoustic waves which are more sensitive to the

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adaptivity of the grid because they are computed with the implicit scheme.

Figure 17 displays the same relative errors versus the gain in CPU time. For the

best error level that can be achieved with the local time stepping algorithms we have a

CPU gain of more than 17, while the gain using the standard multiresolution is around

11.

Eventually we display, in Figure 18, the error between the adaptive solution and

a reference solution obtained using the uniform scheme on a grid corresponding tothe 9th level. The abscissa is the gain in CPU with respect to the uniform solution on

the finest level. As expected, both schemes MR and LTS converge to the error level

corresponding to the error Eu between the uniform solution on the finest level, the 5th

or the 6th, and the reference solution on the 9th level. Compared to the similar curves

with the explicit scheme in Figure 11, the local time stepping enhancement seems to be

more advantageous, knowing that the gas mass fraction is the only component where

precision really matters: all values of threshold tried on the six level hierarchy lead to

a relative error below that obtained with the uniform scheme on the 5th level.

In order to compare all the schemes tested in this study we collect in the follow-

ing table the performances of the six level simulations, in term of computing times,

for both the explicit and implicit schemes. The number in parenthesis are the ratio

between the uniform and the adaptive performances. The gain is significant between

the uniform and adaptive scheme: 9 in the explicit case and 6 in the implicit one.

With the global time stepping enhancement it becomes spectacular: 15 in both case.

The number of calls to the equilibrium state laws is also reported in the table, along

with the computing times. These figures indicate the potential performances of the

algorithms when realistic thermodynamical closure laws will be used instead of the

model one (4.2). One should also note that for the test case presented in this table, we

have used for the explicit scheme a constant time step t = 0.0048 and for the implicit

one t = 0.0325 that is roughly 7 times larger. These values are the minimum values

provided by the stability conditions (2.7) and (2.18) throughout a uniform computa-

tion. The larger time step in the implicit case is counterbalanced by the cost of solving

the linear systems (2.12) and both schemes end up costing roughly the same in termsof computing times in the uniform case.

Explicit Uniform MultiresolutionLocal time

stepping

CPU(s) 31.55 3.54 (8.9) 2.14 (14.7)

State law 10493 1105 (9.5) 237 (44.3)

Semi-implicit Uniform MultiresolutionLocal time

stepping

CPU(s) 33.5 5.8 (5.8) 2.11 (15.9)

State law 17698 2411 (7.3) 345 (51.3)

Table 1 Computing times and number of calls to state laws comparison for

the different algorithms. Threshold parameter = 0.005 and a 6-levelshierarchy.

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6 Conclusion

We have described in details several algorithms to compute the solution of the one

dimensional multiphase flow, in the framework of a LagrangeEuler projection for-

mulation of the equations. Both explicit and semi-implicit schemes are presented, and

for each one, two adaptive enhancement methods are described. The first one is the

standard multiresolution method already implemented in more complicated cases. It

is tested here versus the local time stepping algorithm.

The robustness of the two algorithms has been checked by doing a parameter study.

Several values have been tried of the threshold parameter and for the number of

levels. The relative errors between the different adaptive solutions and the uniform one

present the expected behaviors. The benefits of the local time stepping enhancement

in terms of computing time are encouraging.

Acknowledgments: this work was supported by the Ministere de la Recherche under

grant ERT-20052274: Simulation avancee du transport des hydrocarbures and by the

Institut Francais du Petrole. The authors also wish to thank S. Muller and S.-M. Kaber

for fruitful discussions on the redaction of this paper.

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