a discontinuous galerkin method for stochastic ......numerical methods for classical strong...

A Discontinuous Galerkin Method for

Stochastic Conservation Laws

Yunzhang Li ∗, Chi-Wang Shu †, Shanjian Tang‡

Abstract

In this paper we present a discontinuous Galerkin (DG) method to approximatestochastic conservation laws, which is an efficient high-order scheme. We study the sta-bility for the semi-discrete DG methods for fully nonlinear stochastic equations. Errorestimates are obtained for smooth solutions of semi-linear stochastic equations withvariable coefficients. We also establish a derivative-free second order time discretiza-tion scheme for matrix-valued stochastic ordinary differential equations. Numericalexperiments are performed to confirm the analytical results.

Key Words. Discontinuous Galerkin method, Ito formula, multiplicative stochastic noise,

stability analysis, error estimates, nonlinear stochastic conservation laws, stochastic Burgers

equation.

1 Introduction

Many physical and engineering phenomena may contain some levels of stochastic influences

such as random perturbation of forces and electromagnetic fields, which could be modeled by

stochastic conservation laws. Thus the convergence of numerical methods for the discretiza-

tion of stochastic conservation laws is a topic of high interest. In this paper we present a

discontinuous Galerkin (DG) method for nonlinear stochastic hyperbolic scalar conservation

∗Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shang-hai 200433, China. E-mail: [email protected].†Division of Applied Mathematics, Brown University, RI 02912, USA. E-mail: [email protected].

Research supported by ARO grant W911NF-16-1-0103 and NSF grant DMS-1719410.‡Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shang-

hai 200433, China. E-mail: [email protected]. Research partially supported by National ScienceFoundation of China (Grant No. 11631004) and Science and Technology Commission of Shanghai Munici-pality (Grant No. 14XD1400400).

1

laws with a periodic boundary condition and a multiplicative stochastic perturbation of the

type: du+ f(u)x dt = g(ω, x, t, u) dWt in Ω× [0, 2π]× (0, T ),

u(ω, x, 0) = u0(x), ω ∈ Ω, x ∈ [0, 2π],(1.1)

where the terminal time T > 0 is a fixed real number and Wt, 0 ≤ t ≤ T is a standard

one-dimensional Brownian motion on a given probability space (Ω,F ,P) with a filtration

Ft, 0 ≤ t ≤ T satisfying the usual conditions. We make the following hypotheses:

(H1) The initial condition u0 ∈ L2(0, 2π).

(H2) The function f : R→ R is globally Lipschitz continuous.

(H3) The real scalar function g(ω, x, t, u) is F ⊗ B([0, 2π]) ⊗ B([0, T ]) ⊗ B(R)-measurable.

There exist two constants Lg > 0 and Cg > 0 such that for any (ω, x, t) ∈ Ω× [0, 2π]× [0, T ]

and (u1, u2) ∈ R2,

|g(ω, x, t, u1)− g(ω, x, t, u2)| ≤ Lg|u1 − u2| and |g(ω, x, t, 0)| ≤ Cg.

There are several papers on scalar conservation laws with a multiplicative stochastic

forcing term involving a white noise in time. Feng and Nualart [14] discussed the spatially

one-dimensional case, in which a notion of entropy solution is introduced to prove the exis-

tence and uniqueness of the solution. Later, much effort has been given to extend their results

to the more general spatially multi-dimensional cases and to more extensive initial-boundary

conditions. See e.g. [7, 13, 5, 6, 15]. In this article, we mainly consider the convergence of

numerical methods for classical strong solutions with enough smoothness and integrability.

Concerning the study of numerical schemes for stochastic conservation laws with mul-

tiplicative noises, let us first mention that Bauzet, Charrier and Gallouet proposed several

finite volume schemes. In [2], they studied the convergence of an explicit flux-splitting finite

volume discretization, but with a more restrictive time step stability condition ( ∆t∆x→ 0

as ∆x → 0). Then they investigated the case of a more general flux in [3]. In [4], they

studied the convergence of the scheme when the stochastic conservation law is defined on a

bounded domain with inhomogeneous Dirichlet boundary conditions. Let us also mention

the convergence results of time-discretization of Holden and Risebro [16] and Bauzet [1] on

a bounded domain of Rd, as well as the papers of Kroker [19], and Kroker and Rohde [20]

of finite volume schemes in the one-dimensional case. But none of these articles gives the

order of accuracy for numerical solutions. Also, there seems to be very little attention paid

to the investigation of high-order approximate schemes for stochastic conservation laws.

The DG method we discuss is a class of high-order finite element methods using com-

pletely discontinuous piecewise polynomial space for the numerical solution and the test

functions in the spatial variables, coupled with an explicit and nonlinear stable high-order

time discretization. It was first introduced in 1973 by Reed and Hill [24], in the framework

of neutron transport, which is a deterministic time-independent linear hyperbolic equation.

It was later developed for nonlinear hyperbolic conservation laws containing first derivatives

2

by Cockburn, Shu, et al. in a series of papers [9, 10, 11, 12], in which a framework is given to

efficiently solve deterministic nonlinear time-dependent equations. Since the basis functions

can be discontinuous, the DG methods have certain advantage and flexibility which are not

shared by typical finite element methods such as: (1) it is easy to design high order approxi-

mations, thus allowing for efficient p adaptivity; (2) it is flexible on complicated geometries,

thus allowing for efficient h adaptivity; (3) it is local in data communications, thus allowing

for efficient parallel implementations. In this paper, we shall consider stochastic counterparts

of these works and propose a DG scheme for stochastic conservation laws (1.1) and (3.10),

respectively.

Jiang and Shu [17] proved a cell entropy inequality for the semidiscrete DG method to

possibly nonsmooth solutions of nonlinear conservation laws, which gives the stability result

for the numerical solutions. We shall consider possibly nonsmooth solutions of nonlinear

stochastic equations, and prove that the numerical solutions of the DG scheme are stable.

By similar method, we could also prove the stability of approximate solutions for semilinear

variable-coefficient stochastic conservation laws.

Following the ideas for the deterministic case [25], we give the optimal error estimates

(O(hk+1) for the one-dimensional case) for the semilinear stochastic conservation laws with

variable coefficients. Zhang and Shu [26] presented a priori error estimates for fully discrete

Runge-Kutta DG methods with smooth solutions of scalar nonlinear conservation laws. Un-

fortunately, the unboundedness nature of the stochastic process driven by a Brownian mo-

tion prevents us from applying their method to get the error estimates for the fully nonlinear

stochastic equation.

The DG method is a scheme for spatial discretization, which needs to be coupled with

a high-order time discretization. Unlike the deterministic case, there is no simple heuristic

generalizations of deterministic Runge-Kutta schemes to stochastic differential equations

(SDEs). Kloeden and Platen [18] presented an explicit order 1.5 strong scheme. Milstein

and Tretyakov [23] gave an implementable way to model Ito integrals, which is essential to

construct an order 2.0 (second order) scheme. Combining these methods, in this paper we

establish an explicit order 2.0 strong scheme for SDEs, which seems to be new.

Our effective computational methods for stochastic partial differential equations (SPDEs)

have to face new difficulties. The solutions of SPDEs, when they do exist, are not naturally

time-differentiable, and are not bounded in the path variable. These new features complicate

the calculation and analysis in our stochastic context.

The paper is organized as follows. In Section 2, we introduce notations, definitions and

auxiliary results used in the paper. In Section 3, we present the DG schemes for (1.1)

and (3.10) respectively, and investigate the stability and error estimates of the schemes. In

Section 4, we establish a derivative-free second order time discretization to collaborate with

the semi-discrete scheme presented before. Finally the paper ends with a series of numerical

experiments on some model problems in Section 5 which confirm the analytical results.

3

2 Notations, definitions and auxiliary results

In this section, we introduce notations, definitions, and also some auxiliary results.

2.1 Notations

We denote the mesh by Ij =[xj− 1

2, xj+ 1

2

], for j = 1, ..., N . The center of the cell is

xj = 12

(xj− 1

2+ xj+ 1

2

)and the mesh size is denoted by hj = xj− 1

2− xj+ 1

2, with h = max

1≤j≤Nhj

being the maximum mesh size. We assume that the mesh is regular, namely the ratio between

the maximum and the minimum mesh sizes stays bounded during mesh refinements. We

define the piecewise-polynomial space Vh as the space of polynomials of the degree up to k

in each cell Ij, i.e.

Vh =v : v ∈ P k(Ij) for x ∈ Ij, j = 1, ..., N

.

Note that functions in Vh are allowed to have discontinuities across element interfaces.

We denote by ‖ · ‖ the usual L2(0, 2π) norm with respect to the spatial variable x. The

solution of the numerical scheme is denoted by uh, which belongs to the finite element space

Vh. We denote by (uh)+j+ 1

2and (uh)

−j+ 1

2the values of uh at xj+ 1

2, from the right cell Ij+1,

and from the left cell Ij, respectively. We use the conventional notation [uh] := u+h − u

−h to

denote the jump of the function uh at each element boundary point.

Denote by |x| the Euclidean norm of x ∈ Rk and by 〈x1, x2〉 the inner product of the

vectors x1 and x2 in Rk. An element of Rk×d is a k × d matrix, and its Euclidean norm is

given by |y| :=√trace(yy∗) and the inner product 〈y1, y2〉 := trace(y1y

∗2) for y1, y2 ∈ Rk×d.

By C > 0, we denote a generic constant, which in particular does not depend on the

discretization width h and possibly changes from line to line. Since the Ito integral is not

defined path-wisely, the argument ω of the integrand as a stochastic process will be omitted

in the rest of this paper if there is no danger of confusion.

2.2 The numerical flux

For notational convenience we would like to introduce the following numerical flux related

to the DG spatial discretization. The given monotone numerical flux f (q−, q+) depends on

the two values of the function q at the discontinuity point xj+ 12, namely q±

j+ 12

= q(x±j+ 1

2

).

The numerical flux f (q−, q+) satisfies the following conditions:

(a) it is globally Lipschitz continuous;

(b) it is consistent with the physical flux f(q), i.e., f (q, q) = f(q);

(c) it is nondecreasing in the first argument, and nonincreasing in the second argument.

4

2.3 Inverse property

Finally we list an inverse property of the finite element space Vh that will play a basic role

in our error analysis. There exists a positive constant C such that for any q ∈ Vh,∥∥∥∥∂q∂x∥∥∥∥ ≤ Ch−1 ‖q‖ , (2.1)

where C is independent of q and h. For more details, see Ciarlet [8].

3 The DG method for stochastic conservation laws and

the stability analysis and error estimates

3.1 The DG method for fully nonlinear stochastic conservation

laws

We present the DG method to approximate equation (1.1). For any (ω, t) ∈ Ω× [0, T ], find

uh(ω, ·, t) ∈ Vh such that for any v ∈ Vh,∫Ij

(duh(ω, x, t)) · v(x) dx =

(∫Ij

f (uh(ω, x, t)) vx(x) dx− fj+ 12v−j+ 1

2

+ fj− 12v+j− 1

2

)dt

+

(∫Ij

g (ω, x, t, uh(ω, x, t)) v(x) dx

)dWt, (3.1)

where fj+ 12

:= f(uh(ω, x

−j+ 1

2

, t), uh(ω, x+j+ 1

2

, t))

and v±j+ 1

2

:= v(x±j+ 1

2

) for j = 0, 1, ..., N , and

f is a numerical flux related to the physical flux f .

For x ∈ Ij, the approximating solution should have the form

uh(ω, x, t) =k∑l=0

cjl (ω, t)ϕjl (x),

where ϕjl , l = 0, 1, ..., k is an arbitrary basis of P k(Ij).

Our aim is to solve (3.1) to get cjl (ω, t) : l = 0, 1, ..., k, j = 0, 1, ..., N + 1. Taking

v := ϕjm, m = 0, 1, ..., k, we have

k∑l=0

(∫Ij

ϕjm(x)ϕjl (x) dx

)dcjl (ω, t)

=

(∫Ij

f

(k∑l=0

cjl (ω, t)ϕjl (x)

)ϕjmx(x) dx− fj+ 1

2ϕjm(xj+ 1

2) + fj− 1

2ϕjm(xj− 1

2)

)dt

+

(∫Ij

g

(ω, x, t,

k∑l=0

cjl (ω, t)ϕjl (x)

)ϕjm(x) dx

)dWt.

5

The mass matrix Aj := [Ajml] with

Ajml :=

∫Ij

ϕjm(x)ϕjl (x) dx

is invertible, and its inverse is denoted by Aj,−1.

Then the problem is reduced to solve the following (k+1)×(N+2)-dimensional stochastic

differential equation (SDE):

dc(ω, t) = F (c(ω, t)) dt+G (ω, t, c(ω, t)) dWt, (3.2)

where cl,j(ω, t) := cjl (ω, t),

Fl,j(c) :=

∫Ij

f

(k∑

n=0

cn,j ϕjn(x)

)k∑

m=0

Aj,−1lm ϕjmx(x) dx

−f

(k∑

n=0

cn,j ϕjn(xj+ 1

2),

k∑n=0

cn,j+1 ϕj+1n (xj+ 1

2)

)k∑

m=0

Aj,−1lm ϕjm(xj+ 1

2)

+f

(k∑

n=0

cn,j−1 ϕj−1n (xj− 1

2),

k∑n=0

cn,j ϕjn(xj− 1

2)

)k∑

m=0

Aj,−1lm ϕjm(xj− 1

2),

and

Gl,j (ω, t, c) :=

∫Ij

g

(ω, x, t,

k∑n=0

cn,j ϕjn(x)

)k∑

m=0

Aj,−1lm ϕjm(x) dx.

The initial value of c is determined by u0 as follows:

cjl (0) :=k∑

m=0

Aj,−1lm

∫Ij

u0(x)ϕjm(x) dx. (3.3)

Lemma 3.1. Let Assumptions (H2) and (H3) hold. Then for any N ∈ N+, F and G are

uniformly Lipschitz continuous in the variable c.

Proof. We only show the uniformly Lipschitz continuity of F for fixed N ∈ N, and that of

G can be proved in a similar way.

Fix c, d ∈ R(k+1)×(N+2), l = 0, 1, ..., k, and j = 0, 1, ..., N + 1. We have

Fl,j(c)− Fl,j(d) = El,j + Jl,j +Kl,j,

where

El,j :=

∫Ij

f

(k∑

n=0

cn,j ϕjn(x)

)− f

(k∑

n=0

dn,j ϕjn(x)

)k∑

m=0

Aj,−1lm ϕjmx(x) dx,

6

Jl,j := −f

(k∑

n=0

cn,j ϕjn(xj+ 1

2),

k∑n=0

cn,j+1 ϕj+1n (xj+ 1

2)

)

−f

(k∑

n=0

dn,j ϕjn(xj+ 1

2),

k∑n=0

dn,j+1 ϕj+1n (xj+ 1

2)

) k∑m=0


2),

Kl,j :=

f

(k∑

n=0

cn,j−1 ϕj−1n (xj− 1

2),

k∑n=0

cn,j ϕjn(xj− 1

2)

)

−f

(k∑

n=0

dn,j−1 ϕj−1n (xj− 1

2),

k∑n=0

dn,j ϕjn(xj− 1

2)

) k∑m=0


2).

Since f is globally Lipschitz continuous in the variable c, we have

|El,j| ≤∫Ij

Lf

∣∣∣∣∣k∑

n=0

(cn,j − dn,j) ϕjn(x)

∣∣∣∣∣ ∥∥Aj,−1∥∥∞

k∑m=0

ϕjmx(x) dx

≤

(k∑

n=0

(cn,j − dn,j)2

) 12 ∥∥Aj,−1

∥∥∞

∫Ij

(k∑

n=0

∣∣ϕjn(x)∣∣2) 1

2 k∑m=0

ϕjmx(x) dx

≤ C(h)

(k∑

n=0

(cn,j − dn,j)2

) 12

,

where Lf is the globally Lipschitz constant of f , and C(h) is a positive constant which

depends on h. Then we have

|E|2 =k∑l=0

N+1∑j=0

|El,j|2 ≤k∑l=0

N+1∑j=0

C(h)2

k∑n=0

(cn,j − dn,j)2 = (k + 1)C(h)2(c− d)2.

Since f is globally Lipschitz continuous, we have

|Jl,j| ≤ (k + 1)∥∥Aj,−1

∥∥∞max

m,i

∥∥ϕim∥∥∞ L bf× ∣∣∣∣∣

k∑n=0

(cn,j − dn,j)ϕjn(xj+ 12)

∣∣∣∣∣+

∣∣∣∣∣k∑

n=0

(cn,j+1 − dn,j+1)ϕj+1n (xj+ 1

2)

∣∣∣∣∣

≤ (k + 1)32

∥∥Aj,−1∥∥∞

(maxm,i

∥∥ϕim∥∥∞)2

L bf

×( k∑

n=0

|cn,j − dn,j|2) 1

2

+

(k∑

n=0

|cn,j+1 − dn,j+1|2) 1

2

≤ C(h)

( k∑n=0

|cn,j − dn,j|2) 1

2

+

(k∑

n=0

|cn,j+1 − dn,j+1|2) 1

2 ,

7

where L bf is the global Lipschitz constant of f . Then we have

|J |2 =k∑l=0

N+1∑j=0

|Jl,j|2 ≤k∑l=0

N+1∑j=0

2C(h)2

(k∑

n=0

|cn,j − dn,j|2 +k∑

n=0

|cn,j+1 − dn,j+1|2)

= 4(k + 1)C(h)2 |c− d|2 .

By similar calculation, we could get that

|K|2 ≤ 4(k + 1)C(h)2 |c− d|2 .

Thus

|F (c)− F (d)|2 ≤ 3(|E|2 + |J |2 + |K|2

)≤ C(h) |c− d|2 .

Since u0 is deterministic, by (3.3) we get that c(0) is a deterministic matrix, which is

Lp(Ω)-integrable for any p ≥ 1. Thus, SDE (3.2) has a unique solution c(t)0≤t≤T such that

for any p ≥ 1,

E[

sup0≤t≤T

|c(t)|p]<∞. (3.4)

We have the following stability result for the numerical solutions.

Theorem 3.1. If the assumptions (H1)-(H3) hold, then there exists a constant C > 0 which

is independent of h, such that for any t ∈ [0, T ],

E[‖uh(·, t)‖2] ≤ (C + ‖uh(·, 0)‖2) eCt.

Proof. For any N ∈ N+ and (ω, t) ∈ Ω× [0, T ), take v = uh(ω, ·, t) in (3.1),∫Ij

(duh(x, t)) · uh(x, t) dx =

(∫Ij

f (uh(x, t))uhx(x, t) dx− fj+ 12u−h,j+ 1

2

+ fj− 12u+h,j− 1

2

)dt

+

∫Ij

g (x, t, uh(x, t))uh(x, t) dx dWt. (3.5)

In order to simplify the proof, we use the scaled Legendre basis of P k(Ij) which satisfies

that

Ajml =

∫Ij

ϕjm(x)ϕjl (x) dx = Ajll δml.

Then we have

Aj,−1ml =

1

Ajllδml .

By the orthogonality of the Legendre basis, we get

Gl,j (t, c(t)) = Aj,−1ll

∫Ij

g (x, t, uh(x, t))ϕjl (x) dx.

8

Then for any l ∈ 0, 1, ..., k,

dcjl (t) · dcjl (t) = |Gl,j (t, c(t))|2 dt

=

Aj,−1ll

∫Ij

g (x, t, uh(x, t))ϕjl (x) dx

2

dt

≤(Aj,−1ll

)2∫Ij

|g (x, t, uh(x, t))|2dx∫Ij

∣∣ϕjl (x)∣∣2dx dt

≤ Aj,−1ll

∫Ij

(Cg + Lg|uh(x, t)|)2 dx dt.

Thus we have ∫Ij

duh(x, t) · duh(x, t) dx =

∫Ij

(k∑

m=0

ϕjm(x) dcjm(t)

)2

dx

=k∑l=0

Ajll dcjl (t) · dc

jl (t) ≤ C

∫Ij

(1 + |uh(x, t)|2

)dx dt. (3.6)

According to Ito formula, we have

d |uh(x, t)|2 = 2uh(x, t) duh(x, t) + duh(x, t) · duh(x, t). (3.7)

Combining (3.5), (3.6) and (3.7), we have∫Ij

(d |uh(x, t)|2

)dx

≤ 2

(∫Ij

f (uh(x, t))uhx(x, t) dx− fj+ 12u−h,j+ 1

2

+ fj− 12u+h,j− 1

2

)dt

+2

∫Ij

g (x, t, uh(x, t))uh(x, t) dx dWt + C

∫Ij

(1 + |uh(x, t)|2

)dx dt.

Summarizing over j from 1 to N , we have for t ∈ [0, T ],

‖uh(·, t)‖2 − ‖uh(·, 0)‖2

≤ 2

∫ t

0

N∑j=1

(∫Ij

f (uh(x, s))uhx(x, s) dx− fj+ 12u−h,j+ 1

2

+ fj− 12u+h,j− 1

2

)ds

+2

∫ t

0

∫ 2π

0

g (x, s, uh(x, s))uh(x, s) dx dWs + C + C

∫ t

0

‖uh(·, s)‖2 ds. (3.8)

From (3.4), we have that for any p ≥ 1,

E[∫ T

0

∫ 2π

0

|uh(x, s)|p dx ds]<∞,

9

and thus that the process∫ t

0

∫ 2π

0

g (x, s, uh(x, s))uh(x, s) dx dWs, 0 ≤ t ≤ T

is a martingale. Taking expectation on both sides of inequality (3.8), we have

E[‖uh(·, t)‖2

]≤ ‖uh(·, 0)‖2 + C + C

∫ t

0

E[‖uh(·, s)‖2] ds

+2E

[∫ t

0

N∑j=1

(φ(u−h,j+ 1

2

)− φ

(u+h,j− 1

2

)− fj+ 1

2u−h,j+ 1

2

+ fj− 12u+h,j− 1

2

)ds

]

= C + ‖uh(·, 0)‖2 + C

∫ t

0

E[‖uh(·, s)‖2] ds

+2E

[∫ t

0

N∑j=1

(Fj+ 1

2− Fj− 1

2+ Θj− 1

2

)ds

], (3.9)

where

φ(u) =

∫ u

f(a) da,

Fj+ 12

=(φ(u−h )− fu−h

)j+ 1

2

,

Θj− 12

=(φ(u−h )− φ(u+

h ) + fu+h − fu

−h

)j− 1

2

.

By periodicity, we haveN∑j=1

(Fj+ 1

2− Fj− 1

2

)= 0.

Note that

Θ = φ(u−h )− φ(u+h ) + fu+

h − fu−h

= −φ′(ξ)(u+h − u

−h ) + f

(u+h − u

−h

)=

(f(u−h , u

+h )− f(ξ, ξ)

) (u+h − u

−h

)=

(f(u−h , u

+h )− f(u−h , ξ) + f(u−h , ξ)− f(ξ, ξ)

) (u+h − u

−h

)≤ 0.

Then by (3.9), we have

E[‖uh(·, t)‖2

]≤ C + ‖uh(·, 0)‖2 + C

∫ t

0

E[‖uh(·, s)‖2] ds.

Using Gronwall’s inequality, we have

E[‖uh(·, t)‖2] ≤ (C + ‖uh(·, 0)‖2) eCt.

10

3.2 The DG method for variable-coefficient semi-linear stochastic

conservation laws

The flux f in Equation (1.1) depends only on the variable u. Now we consider a more general

case where the flux depends on u linearly via the variable coefficient a(ω, x, t) as follows: du+ (a(ω, x, t)u)x dt = g(ω, x, t, u) dWt in Ω× [0, 2π]× (0, T ),

u(ω, x, 0) = u0(x), ω ∈ Ω, x ∈ [0, 2π],(3.10)

where a satisfies the following assumption,

(H4) The function a(ω, ·, t) is periodic and smooth for any (ω, t) ∈ Ω × [0, T ] and for any

positive integer l,

sup(ω,x,t)∈Ω×[0,2π]×[0,T ]

|a(ω, x, t)|+

l∑i=1

∣∣∣∣dladxl (ω, x, t)∣∣∣∣<∞.

Analogous to the deterministic case, we present the DG method for variable-coefficient

semi-linear stochastic conservation laws. For any (ω, t) ∈ Ω× [0, T ], find uh(ω, ·, t) ∈ Vh such

that for any v ∈ Vh,∫Ij

(duh(ω, x, t)) · v(x) dx =

(∫Ij

a(ω, x, t)uh(ω, x, t)vx(x) dx− aj+ 12v−j+ 1

2

+ aj− 12v+j− 1

2

)dt

+

(∫Ij

g (ω, x, t, uh(ω, x, t)) v(x) dx

)dWt (3.11)

with

aj+ 12

:= a+(ω, xj+ 12, t)uh(ω, x

−j+ 1

2

, t)− a−(ω, xj+ 12, t)uh(ω, x

+j+ 1

2

, t),

where a+ and a− are the positive and negative parts of the real number a, and thus a =

a+ − a−.

Similar to the above subsection, for x ∈ Ij, the approximating solution should have the

form

uh(ω, x, t) =k∑l=0

cjl (ω, t)ϕjl (x).

We want to solve (3.11) to get cjl (ω, t) : l = 0, 1, ..., k, j = 0, 1, ..., N + 1. Taking

v = ϕjm, m = 0, 1, ..., k, we obtain

k∑l=0

(∫Ij

ϕjm(x)ϕjl (x) dx

)dcjl (ω, t)

=

(∫Ij

a(ω, x, t)

(k∑l=0

cjl (ω, t)ϕjl (x)

)ϕjmx(x) dx− aj+ 1

2ϕjm(xj+ 1

2) + aj− 1

2ϕjm(xj− 1

2)

)dt

11

+

(∫Ij

g

(ω, x, t,

k∑l=0

cjl (ω, t)ϕjl (x)

)ϕjm(x) dx

)dWt.

Then the problem is reduced to solve a (k + 1)× (N + 2)-dimensional SDE as following

dc(ω, t) = F (ω, t, c(ω, t)) dt+G (ω, t, c(ω, t)) dWt, (3.12)

where

cl,j(ω, t) = cjl (ω, t),

Fl,j(ω, t, c) =

∫Ij

a(ω, x, t)

(k∑

n=0

cn,jϕjn(x)

)k∑

m=0

Aj,−1lm ϕjmx(x) dx

−a+(ω, xj+ 12, t)

(k∑

n=0

cn,jϕjn(xj+ 1

2)

)k∑

m=0


2)

+a−(ω, xj+ 12, t)

(k∑

n=0

cn,j+1ϕj+1n (xj+ 1

2)

)k∑

m=0


2)

+a+(ω, xj− 12, t)

(k∑

n=0

cn,j−1ϕj−1n (xj− 1

2)

)k∑

m=0


2)

−a−(ω, xj− 12, t)

(k∑

n=0

cn,jϕjn(xj− 1

2)

)k∑

m=0


2),

and

Gl,j (ω, t, c) =

∫Ij

g

(ω, x, t,

k∑n=0

cn,jϕjn(x)

)k∑

m=0

Aj,−1lm ϕjm(x) dx.

Again we use the L2-projection coefficients of u0 as the initial value of c. That is,

cjl (0) =k∑

m=0

Aj,−1lm

∫Ij

u0(x)ϕjm(x) dx.

Similar to Lemma 3.1, we have from both Hypotheses (H3) and (H4) that F and G are

uniformly Lipschitz-continuous in the variable c. Since u0 is a deterministic function, then

c(0) is a deterministic matrix, which is Lp(Ω)-integrable for any p ≥ 1. Thus according

to classical results of stochastic differential equations, SDE (3.12) has a unique solution

c(t)0≤t≤T such that for any p ≥ 1,

E[

sup0≤t≤T

|c(t)|p]<∞. (3.13)

Similar to Theorem 3.1, we could also obtain that the scheme (3.11) is stable.

12

Theorem 3.2. If the assumptions (H1), (H3) and (H4) hold, then there exists a constant

C > 0, which is independent with h, such that for any t ∈ [0, T ],

E[‖uh(·, t)‖2] ≤ (C + ‖uh(·, 0)‖2) eCt.

Proof. For any N ∈ N+ and (ω, t) ∈ Ω× [0, T ], we define a bilinear functional on piecewisely

smooth function space. For any piecewise smooth functions u, v, define

Hj(a, ω, t;u, v) =

∫Ij

a(ω, x, t)u(x)vx(x) dx

−(a+(ω, xj+ 1

2, t)u−

j+ 12

− a−(ω, xj+ 12, t)u+

j+ 12

)v−j+ 1

2

+(a+(ω, xj− 1

2, t)u−

j− 12

− a−(ω, xj− 12, t)u+

j− 12

)v+j− 1

2

.

Note that∫Ij

a(ω, x, t)u(x)ux(x) dx = −1

2

∫Ij

ax(ω, x, t) |u(x)|2 dx+1

2

[aj+ 1

2

∣∣∣u−j+ 1

2

∣∣∣2 − aj− 12

∣∣∣u+j− 1

2

∣∣∣2] ,where aj+ 1

2= a(ω, xj+ 1

2, t), j = 0, 1, ..., N . Then

Hj(a, ω, t;u, u) = −1

2

∫Ij

ax(ω, x, t) |u(x)|2 dx+1

2

[aj+ 1

2

∣∣∣u−j+ 1

2

∣∣∣2 − aj− 12

∣∣∣u+j− 1

2

∣∣∣2]−aj+ 1

2,+

∣∣∣u−j+ 1

2

∣∣∣2 + aj+ 12,−u

+j+ 1

2

u−j+ 1

2

+ aj− 12,+u−j− 1

2

u+j− 1

2

− aj− 12,−

∣∣∣u+j− 1

2

∣∣∣2= −1

2

∫Ij

ax(ω, x, t) |u(x)|2 dx

+aj+ 12,−u

+j+ 1

2

u−j+ 1

2

− 1

2aj+ 1

2,−

∣∣∣u−j+ 1

2

∣∣∣2 − 1

2aj+ 1

2,+

∣∣∣u−j+ 1

2

∣∣∣2+aj− 1

2,+u

+j− 1

2

u−j− 1

2

− 1

2aj− 1

2,+

∣∣∣u+j− 1

2

∣∣∣2 − 1

2aj− 1

2,−

∣∣∣u+j− 1

2

∣∣∣2= −1

2

∫Ij

ax(ω, x, t) |u(x)|2 dx− Fj+ 12

+ Fj− 12

+aj+ 12,−u

+j+ 1

2

u−j+ 1

2

− 1

2aj+ 1

2,−

∣∣∣u−j+ 1

2

∣∣∣2 − 1

2aj+ 1

2,+

∣∣∣u−j+ 1

2

∣∣∣2+aj+ 1

2,+u

+j+ 1

2

u−j+ 1

2

− 1

2aj+ 1

2,+

∣∣∣u+j+ 1

2

∣∣∣2 − 1

2aj+ 1

2,−

∣∣∣u+j+ 1

2

∣∣∣2= −1

2

∫Ij

ax(ω, x, t) |u(x)|2 dx− Fj+ 12

+ Fj− 12− 1

2

∣∣∣aj+ 12

∣∣∣ · [u]2j+ 12

≤ ‖ax‖∞2

∫Ij

|u(x)|2 dx−(Fj+ 1

2− Fj− 1

2

),

where

Fj+ 12

= aj+ 12,+u

+j+ 1

2

u−j+ 1

2

− 1

2aj+ 1

2,+

∣∣∣u+j+ 1

2

∣∣∣2 − 1

2aj+ 1

2,−

∣∣∣u+j+ 1

2

∣∣∣2 .13

By periodicity we know that

N∑j=1

(Fj+ 1

2− Fj− 1

2

)= 0.

ThusN∑j=1

Hj(a, ω, t;u, u) ≤ 1

2‖ax‖∞ ‖u‖

2 . (3.14)

For any N ∈ N+ and (ω, t) ∈ Ω × [0, T ), take v = uh(ω, ·, t) in (3.11) and do the

summation from j = 1 to j = N ,∫ 2π

0

(duh(x, t)) · uh(x, t) dx =N∑j=1

Hj(a, t;uh(·, t), uh(·, t)) dt

+

∫ 2π

0

g (x, t, uh(x, t))uh(x, t) dx dWt,

≤ 1

2‖ax‖∞ ‖uh(·, t)‖

2 dt+

∫ 2π

0

g (x, t, uh(x, t))uh(x, t) dx dWt.

Similar to the calculation for (3.6) in Theorem 3.1, we have∫ 2π

0

duh(x, t) · duh(x, t) dx ≤ C

∫ 2π

0

(1 + |uh(x, t)|2

)dx dt.

According to Ito formula we have

d |uh(x, t)|2 = 2uh(x, t) duh(x, t) + duh(x, t) · duh(x, t).

Thus∫ 2π

0

(d |uh(x, t)|2

)dx ≤ ‖ax‖∞ ‖uh(·, t)‖

2 dt+ 2

∫ 2π

0

g (x, t, uh(x, t))uh(x, t) dx dWt

+C

∫ 2π

0

(1 + |uh(x, t)|2

)dx dt. (3.15)

By similar arguments in Theorem 3.1 we know that∫ t

0

∫ 2π

0

g (x, s, uh(x, s))uh(x, s) dx dWs, 0 ≤ t ≤ T

is a martingale. Integrating from t = 0 and taking expectation on both sides of (3.15) we

have

E[‖uh(·, t)‖2

]≤ C + ‖uh(·, 0)‖2 + C

∫ t

0

E[‖uh(·, s)‖2] ds.

Lastly Gronwall’s inequality tells us that

E[‖uh(·, t)‖2] ≤ (C + ‖uh(·, 0)‖2) eCt.

Now we state the error estimates of the DG method (3.11).

14

Theorem 3.3. Suppose that assumptions (H1), (H3) and (H4) hold and equation (3.10) has

a unique strong solution u(·) such that

(H5) u(·) ∈ L2(Ω× [0, T ];Hk+2

)⋂L4 (Ω× [0, 2π]× [0, T ]; R)

⋂L∞(0, T ;L2(Ω;Hk+1)

);

(H6) g (·, u(·)) ∈ L2(Ω× [0, T ];Hk+1

).

Then there exists a constant C > 0, which is independent with h, such that for any t ∈ [0, T ],(E[‖u(·, t)− uh(·, t)‖2]) 1

2 ≤ CeCthk+1.

Proof. Notice that the scheme (3.11) is also satisfied when the numerical solution uh(·) is

replaced by the exact solution u(·), for any v ∈ Vh,∫Ij

(du(ω, x, t)) · v(x) dx =

(∫Ij

a(ω, x, t)u(ω, x, t)vx(x) dx− auj+ 12v−j+ 1

2

+ auj− 12v+j− 1

2

)dt

+

(∫Ij

g (ω, x, t, u(ω, x, t)) v(x) dx

)dWt, (3.16)

with

auj+ 12

= a+(ω, xj+ 12, t)u(ω, x−

j+ 12

, t)− a−(ω, xj+ 12, t)u(ω, x+

j+ 12

, t).

Define

e(ω, x, t) = u(ω, x, t)− uh(ω, x, t) = ξ(ω, x, t)− η(ω, x, t),

with

ξ(ω, x, t) = Pu(ω, x, t)− uh(ω, x, t), η(ω, x, t) = Pu(ω, x, t)− u(ω, x, t),

where P is a projection from Hk+1 onto Vh, which will be specified later.

By (3.11) and (3.16), we have the error equation∫Ij

(de(ω, x, t)) · v(x) dx

=

(∫Ij

a(ω, x, t)e(ω, x, t)vx(x) dx− aej+ 12v−j+ 1

2

+ aej− 12v+j− 1

2

)dt

+

∫Ij

g (ω, x, t, u(ω, x, t))− g (ω, x, t, uh(ω, x, t))

v(x) dx dWt,

with

aej+ 12

= a+(ω, xj+ 12, t)e(ω, x−

j+ 12

, t)− a−(ω, xj+ 12, t)e(ω, x+

j+ 12

, t).

It turns out that∫Ij

(dξ(x, t)) · v(x) dx

=

(∫Ij

(dη(x, t)) · v(x) dx+Hj(a, ω, t; ξ(·, t), v)−Hj(a, ω, t; η(·, t), v)

)dt

15

+

∫Ij

g (x, t, u(x, t)))− g (x, t, uh(x, t))

v(x) dx dWt.

Taking v = ξ(·, t) and doing the summation from j = 1 to j = N we have∫ 2π

0

ξ(x, t)dξ(x, t) dx

=

∫ 2π

0

ξ(x, t)dη(x, t) dx+N∑j=1

(Hj(a, ω, t; ξ(·, t), ξ(·, t))−Hj(a, ω, t; η(·, t), ξ(·, t))

)dt

+

∫ 2π

0

g (x, t, u(x, t)))− g (x, t, uh(x, t))

ξ(x, t) dx dWt.

According to Ito formula we get

d |ξ(x, t)|2 = 2ξ(x, t) dξ(x, t) + dξ(x, t) · dξ(x, t).

Then we have

E[‖ξ(·, t)‖2] = ‖ξ(·, 0)‖2 + T1(t) + T2(t) + T3(t) + T4(t) + T5(t) ,

where

T1(t) = −2E

[∫ t

0

N∑j=1

Hj(a, ω, s; η(·, s), ξ(·, s)) ds

],

T2(t) = E[∫ 2π

0

∫ t

0

dξ(x, s) · dξ(x, s) dx],

T3(t) = 2E[∫ 2π

0

∫ t

0

ξ(x, s)dη(x, s) dx

],

T4(t) = 2E

[∫ t

0

N∑j=1

Hj(a, ω, s; ξ(·, s), ξ(·, s)) ds

],

T5(t) = 2E[∫ t

0

∫ 2π

0

g (x, s, u(x, s)))− g (x, s, uh(x, s))

ξ(x, s) dx dWs

]will be estimated separately later.

• T1(t) term.

We define the projection P on piecewise smooth function space as the following way. For

any fixed (ω, t) ∈ Ω× [0, T ], j ∈ 1, 2, ..., N and piecewise smooth function u,

Case 1: If a(ω, xj− 12, t) < 0 and a(ω, xj+ 1

2, t) > 0,

∫Ij

(Pu− u) (x) · v(x) dx = 0, ∀v ∈ P k−2(Ij)

(Pu− u)−j+ 1

2= 0,

(Pu− u)+j− 1

2= 0.

16

Case 2: If a(ω, xj− 12, t) > 0 and a(ω, xj+ 1

2, t) > 0,

∫Ij

(Pu− u) (x) · v(x) dx = 0, ∀v ∈ P k−1(Ij)

(Pu− u)−j+ 1

2= 0.

Case 3: If a(ω, xj− 12, t) < 0 and a(ω, xj+ 1

2, t) < 0,

∫Ij

(Pu− u) (x) · v(x) dx = 0, ∀v ∈ P k−1(Ij)

(Pu− u)+j− 1

2= 0.

Case 4: If a(ω, xj− 12, t) > 0 and a(ω, xj+ 1

2, t) < 0,∫

Ij

(Pu− u) (x) · v(x) dx = 0, ∀v ∈ P k(Ij).

Then according to the classical projection theory and (H4), we know that there is a

constant C > 0 that is independent with ω, t, u and h, such that

‖u− Pu‖ ≤ C ‖u‖Hk+1 hk+1. (3.17)

Note that

Hj(a, ω, t; η(·, s), ξ(·, s)) =

∫Ij

a(ω, x, t)η(ω, x, s)ξx(ω, x, s) dx

−(a+(ω, xj+ 1

2, t)η−

j+ 12

− a−(ω, xj+ 12, t)η+

j+ 12

)ξ−j+ 1

2

+(a+(ω, xj− 1

2, t)η−

j− 12

− a−(ω, xj− 12, t)η+

j− 12

)ξ+j− 1

2

.

By the properties of the projection P , we can verify that for all j ∈ 1, 2, ..., N

−(a+(ω, xj+ 1

2, t)η−

j+ 12

− a−(ω, xj+ 12, t)η+

j+ 12

)ξ−j+ 1

2

+(a+(ω, xj− 1

2, t)η−

j− 12

− a−(ω, xj− 12, t)η+

j− 12

)ξ+j− 1

2

= 0.

For the term∫Ija(ω, x, t)η(ω, x, s)ξx(ω, x, s) dx, we study it case by case.

Case 1 & Case 4: Since a(ω, xj− 12, t) · a(ω, xj+ 1

2, t) < 0, there must exist yj ∈ Ij such that

a(ω, yj, t) = 0. Then according to (H4) we have

|a(ω, x, t)| = |a(ω, yj, t) + ax(ω, ξ1, t)(x− yj)| ≤ Ch.

By inverse inequality (2.1), we have∣∣∣∣∣∫Ij


∣∣∣∣∣17

≤ Ch

∫Ij

|η(ω, x, t)ξx(ω, x, t)| dx

≤ Ch‖η(ω, ·, t)‖Ij‖ξx(ω, ·, t)‖Ij ≤ Ch‖η(ω, ·, t)‖IjCh−1‖ξ(ω, ·, t)‖Ij≤ C‖η(·, t)‖2

Ij+ C‖ξ(·, t)‖2

Ij.

Case 2 & Case 3: Note that

a(ω, x, t) = a(ω, xj, t) + ax(ω, ξ2, t)(x− xj).

It follows that∣∣∣∣∣∫Ij


∣∣∣∣∣≤

∣∣∣∣∣a(ω, xj, t)

∫Ij

η(ω, x, s)ξx(ω, x, s) dx

∣∣∣∣∣+ Ch

∫Ij

|η(ω, x, t)ξx(ω, x, t)| dx

≤ Ch‖η(ω, ·, t)‖Ij‖ξx(ω, ·, t)‖Ij ≤ Ch‖η(ω, ·, t)‖IjCh−1‖ξ(ω, ·, t)‖Ij≤ C‖η(·, t)‖2

Ij+ C‖ξ(·, t)‖2

Ij.

Then we have from (H5) and (3.17),

T1(t) = −2E

[∫ t

0

N∑j=1

Hj(a, ω, s; η(·, s), ξ(·, s)) ds

]

≤ CE[∫ t

0

‖η(·, s)‖2 ds

]+ CE

[∫ t

0

‖ξ(·, s)‖2 ds

]≤ CE

[∫ t

0

‖u(·, s)‖2Hk+1 ds

]h2k+2 + CE

[∫ t

0

‖ξ(·, s)‖2 ds

]≤ Ch2k+2 + C

∫ t

0

E[‖ξ(·, s)‖2

]ds.

• T2(t) term.

Since

dt(Pu)(ω, x, t) = P(dtu)(ω, x, t)

= −P((au)x

)(ω, x, t) dt+ P

(g(ω, ·, t, u(ω, ·, t))

)(x) dWt, (3.18)

we have ∫Ij

(dPu(ω, x, t)) · v(x) dx

=

(∫Ij

−P((au)x

)(ω, x, t) · v(x) dx

)dt

18

+

(∫Ij

P(g (ω, ·, t, u(ω, ·, t))

)(x) · v(x) dx

)dWt. (3.19)

From (3.11) and (3.19), we have∫Ij

(dξ(x, t)) · v(x) dx

=

(∫Ij

−P((au)x

)(ω, x, t) · v(x) dx−Hj(a, ω, t;uh(·, t), v)

)dt

+

(∫Ij

(P(g (·, t, u(·, t))

)− g (·, t, uh(·, t))

)(x) · v(x) dx

)dWt.

Since ξ(ω, ·, t) ∈ Vh for any (ω, t) ∈ Ω× [0, T ] , then ξ should have the form

ξ(ω, x, t) =k∑l=0

ξjl (ω, t)ϕjl (x), x ∈ Ij.

Using the scaled Legendre basis, similar to the arguments in Theorem 3.1, we have for

l = 0, 1, ..., k,

dξjl (t) · dξjl (t) =

Aj,−1ll

∫Ij

(P(g (·, t, u(·, t))

)− g (·, t, uh(·, t))

)(x) · ϕjl (x) dx

2

dt

≤ 2(Aj,−1ll

)2∫Ij

∣∣P(g (·, t, u(·, t)))− g (·, t, u(·, t))

∣∣2 (x) dx

∫Ij


+2(Aj,−1ll

)2∫Ij

|g (x, t, u(x, t))− g (x, t, uh(x, t))|2 dx∫Ij


≤ 2Aj,−1ll

∫Ij

∣∣P(g (·, t, u(·, t)))− g (·, t, u(·, t))

∣∣2 (x) dx dt

+2L2gA

j,−1ll

∫Ij

|u(x, t)− uh(x, t)|2 dx dt

≤ CAj,−1ll

∫Ij

∣∣P(g (·, t, u(·, t)))− g (·, t, u(·, t))

∣∣2 (x) dx dt

+CAj,−1ll

∫Ij

|η(x, t)|2 dx dt+ CAj,−1ll

∫Ij

|ξ(x, t)|2 dx dt.

Then we have∫Ij

dξ(x, t) · dξ(x, t) dx =

∫Ij

(k∑

m=0

ϕjm(x) dξjm(t)

)2

dx =k∑l=0

Ajll dξjl (t) · dξ

jl (t)

≤ C

∫Ij

∣∣P(g (·, t, u(·, t)))− g (·, t, u(·, t))

∣∣2 (x) dx dt

19

+C

∫Ij

|η(x, t)|2 dx dt+ C

∫Ij

|ξ(x, t)|2 dx dt.

It follows that∫ 2π

0

dξ(x, t) · dξ(x, t) dx

≤ C‖Pg(·, t, u(·, t))− g(·, t, u(·, t))‖2 dt+ C‖Pu(·, t)− u(·, t)‖2 dt+ C‖ξ(·, t)‖2 dt

≤ C(‖g(·, t, u(·, t))‖2

Hk+1 + ‖u(·, t)‖2Hk+1

)h2k+2 dt+ C‖ξ(·, t)‖2 dt.

Then by (H5) and (H6) we get

T2(t) = E[∫ 2π

0

∫ t

0

dξ(x, s) · dξ(x, s) dx]≤ Ch2k+2 + C

∫ t

0

E[‖ξ(·, s)‖2

]ds.

• T3(t) term.

By (3.10) and (3.18), we have

dη(ω, x, t) =

(au)x (ω, x, t)− P(

(au)x)(ω, x, t)

dt

+P(g(ω, ·, t, u(ω, ·, t))

)(x)− g(ω, x, t, u(ω, x, t))

dWt.

Thus ∫ 2π

0

∫ t

0


=

∫ 2π

0

∫ t

0

(au)x (x, s)− P

((au)x

)(x, s)

ξ(x, s)dsdx

+

∫ t

0

∫ 2π

0

P(g(·, t, u(·, t))

)(x)− g(x, s, u(x, s))

ξ(x, s) dxdWs.

According to (3.13) and u ∈ L4 (Ω× [0, 2π]× [0, T ]; R), we know that the process∫ t

0

∫ 2π

0

P(g(·, t, u(·, t))

)(x)− g(x, s, u(x, s))

ξ(x, s) dxdWs, 0 ≤ t ≤ T

is a martingale. Then

T3(t) = 2E[∫ 2π

0

∫ t

0


]= 2E

[∫ 2π

0

∫ t

0

(au)x (x, s)− P

((au)x

)(x, s)

ξ(x, s)dsdx

]≤ E

[∫ t

0

∫ 2π

0

∣∣(au)x (x, s)− P(

(au)x)(x, s)

∣∣2 dxds]+

∫ t

0

E[‖ξ(·, s)‖2

]ds

≤ CE[∫ t

0

‖(au)x(·, s)‖2Hk+1 ds

]h2k+2 +

∫ t

0

E[‖ξ(·, s)‖2

]ds.

20

Since

‖(au)x(·, s)‖Hk+1 ≤ C‖(au)(·, s)‖Hk+2 ≤ C‖u(·, s)‖Hk+2

and u(·) ∈ L2(Ω× [0, T ];Hk+2

), we get

T3(t) ≤ Ch2k+2 + C

∫ t

0

E[‖ξ(·, s)‖2

]ds.

• T4(t) term.

According to (3.14), we get

T4(t) = 2E

[∫ t

0

N∑j=1

Hj(a, ω, s; ξ(·, s), ξ(·, s)) ds

]

≤ ‖ax‖∞∫ t

0

E[‖ξ(·, s)‖2

]ds.

• T5(t) term.

By virtue of (H5) and (3.13), we know that the process∫ t

0

∫ 2π

0

(g (x, s, u(x, s)))− g (x, s, uh(x, s))

)ξ(x, s) dxdWs, 0 ≤ t ≤ T

is a martingale. Then

T5(t) = 2E[∫ t

0

∫ 2π

0

(g (x, s, u(x, s)))− g (x, s, uh(x, s))

)ξ(x, s) dxdWs

]= 0.

Concluding the above arguments, we have

E[‖ξ(·, t)‖2

]≤ Ch2k+2 + C

∫ t

0

E[‖ξ(·, s)‖2

]ds.

Using Gronwall’s inequality, we have(E[‖ξ(·, t)‖2

]) 12 ≤ Chk+1eCt.

According to (3.17) and u ∈ L∞(0, T ;L2(Ω;Hk+1)

), we have(

E[‖η(·, t)‖2

]) 12 ≤ C

(E[‖u(·, t)‖2

Hk+1

]) 12 hk+1 ≤ Chk+1.

It turns out that(E[‖u(·, t)− uh(·, t)‖2]) 1

2 ≤(E[‖ξ(·, t)‖2

]) 12 +

(E[‖η(·, t)‖2

]) 12 ≤ CeCthk+1.

Remark 3.1. The solution of stochastic conservation law usually does not have a uniform

bound with respect to the variable ω ∈ Ω. Thus we could not generalize the method in Zhang

and Shu [26] to get the error estimates for fully nonlinear stochastic conservation law, in

which they made use of the uniform boundedness of the approximate solutions. But inter-

estingly, numerical examples in Section 5.3 verify the optimal order O(hk+1) for nonlinear

stochastic equations.

21

4 Time discretization

The DG method only involves the spatial discretization, and transfers the primal stochastic

partial differential equation into a SDE. Thus we need to derive an implementable high-

order time discretization. For notational simplicity, we shall mainly state the schemes for

the autonomous case. Consider the following matrix-valued SDE: dX i,jt = ai,j(Xt) dt+ bi,j(Xt) dWt, t > 0

X i,j0 = xi,j0 ,

where i = 0, 1, ..., k and j = 0, 1, ..., N + 1. We aim to use Y i,jn to approximate X i,j

tn . Define

Y i,j0 := xi,j0 . Suppose we already have Y i,j

n : i = 0, 1, ..., k and j = 0, 1, ..., N + 1.

4.1 Taylor order 2.0 strong scheme

Define the following operators

L0f :=N+1∑j=0

k∑i=0

ai,j∂f

∂xij+

1

2

N+1∑l,j=0

k∑m,i=0

bi,jbm,l∂2f

∂xij∂xml,

and

L1f :=N+1∑j=0

k∑i=0

bi,j∂f

∂xij,

where f : R(k+1)×(N+2) −→ R is twice differentiable.

According to [18, Theorem 11.5.1, page 391], the order 2.0 strong Taylor scheme is

Y i,jn+1 = Y i,j

n + ai,j(Yn)(tn+1 − tn) + bi,j(Yn)(Wtn+1 −Wtn)

(order 0.5)

+L1bi,j(Yn)

∫ tn+1

tn

∫ s

tn

dWrdWs

(order 1.0)

+1

2L0ai,j(Yn) (tn+1 − tn)2 + L0bi,j(Yn)

∫ tn+1

tn

∫ s

tn

drdWs

+L1ai,j(Yn)

∫ tn+1

tn

∫ s

tn

dWrds+ L1L1bi,j(Yn)

∫ tn+1

tn

∫ s

tn

∫ r

tn

dWudWrdWs

(order 1.5)

+L1L0bi,j(Yn)

∫ tn+1

tn

∫ s

tn

∫ r

tn

dWudrdWs

+L1L1ai,j(Yn)

∫ tn+1

tn

∫ s

tn

∫ r

tn

dWudWrds

22

+L0L1bi,j(Yn)

∫ tn+1

tn

∫ s

tn

∫ r

tn

dudWrdWs

+L1L1L1bi,j(Yn)

∫ tn+1

tn

∫ s

tn

∫ r

tn

∫ u

tn

dWvdWudWrdWs.

(order 2.0) (4.1)

Define

∆n = tn+1 − tn, ∆Wn = Wtn+1 −Wtn ,

and

∆Zn =

∫ tn+1

tn

(Ws −Wtn) ds, ∆Un =

∫ tn+1

tn

(Ws −Wtn)2 ds.

By Ito formula we have∫ tn+1

tn

∫ s

tn

dWrdWs =1

2

(∆Wn)2 −∆n

,

∫ tn+1

tn

∫ s

tn

drdWs = ∆Wn∆n −∆Zn,∫ tn+1

tn

∫ s

tn

∫ r

tn

dWudWrdWs =1

6

(∆Wn)2 − 3∆n

∆Wn,∫ tn+1

tn

∫ s

tn

∫ r

tn

dWudrdWs = −∆Un + ∆Wn∆Zn,∫ tn+1

tn

∫ s

tn

∫ r

tn

dWudWrds =1

2∆Un −

1

4∆2n,∫ tn+1

tn

∫ s

tn

∫ r

tn

dudWrdWs =1

2∆Un −∆Wn∆Zn +

1

2(∆Wn)2 ∆n −

1

4∆2n,∫ tn+1

tn

∫ s

tn

∫ r

tn

∫ u

tn

dWvdWudWrdWs =1

24

(∆Wn)4 − 6 (∆Wn)2 ∆n + 3∆2

n

.

Thus we could rewrite the Taylor scheme (4.1) as follows,

Y i,jn+1 = Y i,j

n + ai,j(Yn)∆n + bi,j(Yn)∆Wn

(order 0.5)

+1

2L1bi,j(Yn)

(∆Wn)2 −∆n

(order 1.0)

+1

2L0ai,j(Yn)∆2

n + L0bi,j(Yn) ∆Wn∆n −∆Zn

+L1ai,j(Yn)∆Zn +1

6L1L1bi,j(Yn)

(∆Wn)2 − 3∆n

∆Wn

(order 1.5)

23

+L1L0bi,j(Yn) −∆Un + ∆Wn∆Zn+ L1L1ai,j(Yn)

1

2∆Un −

1

4∆2n

+L0L1bi,j(Yn)

1


1

2(∆Wn)2 ∆n −

1

4∆2n

+

1

24L1L1L1bi,j(Yn)

(∆Wn)4 − 6 (∆Wn)2 ∆n + 3∆2

n

,

(order 2.0) (4.2)

where the method of modeling the stochastic variables ∆Wn, ∆Zn and ∆Un will be specified

later.

4.2 Explicit order 2.0 strong scheme

A disadvantage of the strong Taylor approximations is that the derivatives of various or-

ders of the drift and diffusion coefficients must be evaluated at each step, in addition to

the coefficients themselves. This can make implementation of such schemes a complicated

undertaking. In this subsection we will propose a strong scheme which avoids the usage of

derivatives in much the same way that Runge-Kutta schemes do in the deterministic setting.

4.2.1 Derivative-free scheme

Following the idea of [18], we could derive a derivative-free scheme of order 2.0 by replacing

the derivatives in the order 2.0 strong Taylor scheme (4.2) by corresponding finite differences.

We set

γm,l± = Y m,ln + am,l(Yn)∆n ± bm,l(Yn)

√∆n,

ηm,l± = Y m,ln ± bm,l(Yn)∆n;

φm,l+,± = γm,l+ + am,l(γ+)∆n ± bm,l(γ+)√

∆n,

φm,l−,± = γm,l− + am,l(γ−)∆n ± bm,l(γ−)√

∆n;

βm,l+,± = φm,l+,+ ± bm,l(φ+,+)√

∆n,

βm,l−,± = φm,l+,− ± bm,l(φ+,−)√

∆n. (4.3)

One could easily verify that

L1bi,j(Yn) =1

2∆n

bi,j(η+)− bi,j(η−)

+O(∆2

n),

L0ai,j(Yn) =1

2∆n

ai,j(γ+)− 2ai,j(Yn) + ai,j(γ−)

+O(∆n),

24

L0bi,j(Yn) =1

2∆n

bi,j(γ+)− 2bi,j(Yn) + bi,j(γ−)

+O(∆n),

L1ai,j(Yn) =1

2√

∆n

ai,j(γ+)− ai,j(γ−)

+O(∆n),

L1L1bi,j(Yn) =1

4∆n

bi,j(φ+,+)− bi,j(φ+,−)− bi,j(φ−,+) + bi,j(φ−,−)

+O(∆n),

L1L0bi,j(Yn) =1

2∆32n

bi,j(φ+,+) + bi,j(φ+,−)− 3bi,j(γ+)

−bi,j(γ−) + 2bi,j(Yn)

+O(

√∆n),

L1L1ai,j(Yn) =1

2∆n

ai,j(φ+,+)− ai,j(φ+,−)− ai,j(γ+) + ai,j(γ−)

+O(

√∆n),

L0L1bi,j(Yn) =1

4∆32n

bi,j(φ+,+)− bi,j(φ+,−) + bi,j(φ−,+)− bi,j(φ−,−)

−2bi,j(γ+) + 2bi,j(γ−)

+O(

√∆n),

L1L1L1bi,j(Yn) =1

4∆32n

bi,j(β+,+)− bi,j(β+,−)− bi,j(β−,+) + bi,j(β−,−)− bi,j(φ+,+)

+bi,j(φ+,−) + bi,j(φ−,+)− bi,j(φ−,−)

+O(

√∆n).

Then we could rewrite scheme (4.2) as the following scheme

Y i,jn+1 = Y i,j

n + ai,j(Yn)∆n + bi,j(Yn)∆Wn

(order 0.5)

+1

4∆n


(∆Wn)2 −∆n

(order 1.0)

+1

4


∆n

+1

2√

∆n


∆Zn

+1

2∆n


∆Wn∆n −∆Zn

25

+1

8∆n


1

3(∆Wn)2 −∆n

∆Wn

(order 1.5)

+1

2∆32n

bi,j(φ+,+) + bi,j(φ+,−)− 3bi,j(γ+)− bi,j(γ−) + 2bi,j(Yn)

−∆Un + ∆Wn∆Zn

+1

2∆n


1

2∆Un −

1

4∆2n

+

1

4∆32n

bi,j(φ+,+)− bi,j(φ+,−) + bi,j(φ−,+)− bi,j(φ−,−)− 2bi,j(γ+) + 2bi,j(γ−)

×

1


1

2(∆Wn)2 ∆n −

1

4∆2n

+

1

96∆32n

bi,j(β+,+)− bi,j(β+,−)− bi,j(β−,+) + bi,j(β−,−)− bi,j(φ+,+) + bi,j(φ+,−)

+bi,j(φ−,+)− bi,j(φ−,−)

×

(∆Wn)4 − 6 (∆Wn)2 ∆n + 3∆2n

.

(order 2.0) (4.4)

4.2.2 Modeling of Ito integrals

We have proposed a derivative-free scheme (4.4). Now it remains to model at each step three

random variables ∆Wn, ∆Zn and ∆Un. In [22], the characteristic function of these random

variables is found. However, it is very complicated and cannot be useful in practice. Thus,

the exact modeling has bad perspectives, and therefore we need to be able to model these

variables approximately. The method of modeling can be found in [23]. For convenience of

the reader, we give a full detailed description of the modeling algorithm here.

Introduce the new process

v(s) =Wtn+∆ns −Wtn√

∆n

, 0 ≤ s ≤ 1.

It is obvious that v(s), 0 ≤ s ≤ 1 is a standard Wiener process. We have

∆Wn = ∆12nv(1), ∆Zn = ∆

32n

∫ 1

0

v(s) ds, ∆Un = ∆2n

∫ 1

0

v2(s) ds.

Then the problem of modeling the random variables ∆Wn, ∆Zn and ∆Un could be

reduced to that of modeling the variables v(1),∫ 1

0v(s) ds and

∫ 1

0v2(s) ds. These variables

are the solution of the system of equationsdx = dv(s), x(0) = 0,

dy = x ds, y(0) = 0,

dz = x2 ds, z(0) = 0,

(4.5)

at the moment s = 1.

26

Let xk = x(sk), yk = y(sk), zk = z(sk), 0 = s0 < s1 < · · · < sNn = 1, sk+1− sk = δ = 1Nn

,

be an approximate solution of (4.5), where Nn is to be determined. We will now use a

method of order 1.5 to integrate (4.5).xk+1 = xk + (v(sk+1)− v(sk)),

yk+1 = yk + xkδ +

∫ sk+1

sk

(v(θ)− v(sk)) dθ,

zk+1 = zk + x2kδ + 2xk

∫ sk+1

sk

(v(θ)− v(sk)) dθ +δ2

2.

(4.6)

Here the additional random variable∫ sk+1

sk(v(θ)−v(sk)) dθ is normally distributed with mean,

variance and correlation

E[∫ sk+1

sk

(v(θ)− v(sk)) dθ

]= 0,

E

[(∫ sk+1

sk


)2]

=1

3δ3,

E[(v(sk+1)− v(sk)

)·(∫ sk+1

sk


)]=

1

2δ2,

respectively. We note that there is no difficulty in generating the pair of correlated normally

distributed random variables v(sk+1) − v(sk) and∫ sk+1

sk(v(θ) − v(sk)) dθ using the transfor-

mation

v(sk+1)− v(sk) = ζk,1δ12 ,

∫ sk+1

sk

(v(θ)− v(sk)) dθ =1

2

(ζk,1 +

1√3ζk,2

)δ

32 , (4.7)

where ζk,1 and ζk,2 are independent normally N(0; 1) distributed random variables.

The method (4.6) has the following properties. Firstly xk and yk are equal to v(sk) and∫ sk

0v(θ) dθ exactly. Secondly we have(

E

[∣∣∣∣zNn −∫ 1

0

v2(s) ds

∣∣∣∣2]) 1

2

= O(δ32 ).

We choose δ such that δ = O(∆13n ) i.e.

Nn =

⌈∆− 1

3n

⌉, (4.8)

with d·e standing for the ceiling function.

Then we have ∆12nxNn = ∆Wn, ∆

32nyNn = ∆Zn and(

E[∣∣∆2

nzNn −∆Un∣∣2]) 1

2= O(∆

52n ).

Thus according to [23, Theorem 4.2, page 50], in a method of second order of accuracy

with time step ∆n such as scheme (4.4), we could replace ∆Wn, ∆Zn and ∆Un by ∆12nxNn ,

27

∆32nyNn and ∆2

nzNn independently at each step. Finally, we get an implementable derivative-

free order 2.0 time discretization scheme,

Y i,jn+1 = Y i,j

n + ai,j(Yn)∆n + bi,j(Yn)xNn

√∆n

(order 0.5)

+1

4


x2Nn− 1

(order 1.0)

+1

4


∆n +

1

2


yNn∆n

+1

2


xNn − yNn

√∆n

+1

8


1

3x2Nn− 1

xNn

√∆n

(order 1.5)

+1

2

bi,j(φ+,+) + bi,j(φ+,−)− 3bi,j(γ+)− bi,j(γ−) + 2bi,j(Yn)

xNnyNn − zNn

√∆n

+1

2


1

2zNn −

1

4

∆n

+1

4

bi,j(φ+,+)− bi,j(φ+,−) + bi,j(φ−,+)− bi,j(φ−,−)− 2bi,j(γ+) + 2bi,j(γ−)

×

1

2zNn − xNnyNn +

1

2x2Nn− 1

4

√∆n

+1

96

bi,j(β+,+)− bi,j(β+,−)− bi,j(β−,+) + bi,j(β−,−)− bi,j(φ+,+) + bi,j(φ+,−)

+bi,j(φ−,+)− bi,j(φ−,−)

×x4Nn− 6x2

Nn+ 3√

∆n,

(order 2.0) (4.9)

where xNn , yNn , zNn are computed by (4.6), (4.7), (4.8), and γ±, η±, φ±,±, β±,± are calculated

by (4.3).

5 Numerical experiments

In this section we consider the application of the numerical methods, which we defined in

section 3 and section 4, on some model problems. Here, M is the number of realizations of

the stochastic approximate solutions. We use the average of M realizations to approximate

the mathematical expectation. The degree of the piecewise-polynomial space Vh is k. The

positive real number T is the terminal time. In all experiments of DG scheme with k = 1,

we set ∆t = ∆x2k+1

so that scheme (4.9) is efficiently second-order and the CFL condition is

satisfied. In all experiments of DG scheme with k = 2, we have adjusted the time step to

∆t ∼ (∆x)32 so that the scheme in time is effectively third-order.

28

5.1 Constant-Coefficient linear stochastic equation

We consider the following linear equation du+ ux dt = bu dWt in Ω× [0, 2π]× (0, T ),

u(ω, x, 0) = u0(x), ω ∈ Ω, x ∈ [0, 2π].(5.1)

The exact solution of (5.1) is

u(ω, x, t) = u0(x− t)ebWt(ω)− 12b2t.

The numerical flux is taken as the simple upwind flux f (u−, u+) = u−. In Table 1 and

Table 2, we show the errors of DG scheme (3.1) with 10000 realizations and u0(x) = sin(x).

Table 1: Accuracy on (5.1) with k = 1, M = 10000, u0(x) = sin(x)

b = 0.1 b = 0.5 b = 1.0

N L2 Error order L2 Error order L2 Error order

T = 0.5

10 4.12E-02 - 4.38E-02 - 5.28E-02 -

20 1.06E-02 1.96 1.12E-02 1.96 1.35E-02 1.96

40 2.66E-03 1.99 2.82E-03 1.99 3.39E-03 2.00

80 6.67E-04 2.00 7.08E-04 2.00 8.50E-04 2.00

160 1.67E-04 2.00 1.78E-04 1.99 2.18E-04 1.96

320 4.17E-05 2.00 4.42E-05 2.01 5.28E-05 2.05

T = 1.0

10 4.15E-02 - 4.68E-02 - 6.70E-02 -

20 1.06E-02 1.97 1.21E-02 1.96 1.79E-02 1.91

40 2.67E-03 1.99 3.01E-03 2.00 4.31E-03 2.05

80 6.68E-04 2.00 7.48E-04 2.01 1.04E-03 2.05

160 1.67E-04 2.00 1.90E-04 1.98 2.72E-04 1.93

320 4.18E-05 2.00 4.68E-05 2.02 6.76E-05 2.01

T = 2.0

10 4.29E-02 - 5.41E-02 - 1.09E-01 -

20 1.08E-02 1.99 1.38E-02 1.97 2.97E-02 1.87

40 2.69E-03 2.00 3.42E-03 2.01 7.56E-03 1.97

80 6.73E-04 2.00 8.59E-04 1.99 1.91E-03 1.98

160 1.69E-04 2.00 2.17E-04 1.99 4.36E-04 2.14

320 4.21E-05 2.00 5.39E-05 2.01 1.19E-04 1.87

From Table 1 and Table 2, we see that the order of accuracy of the DG scheme (3.1) is

k + 1, which is consistent with the result in Theorem 3.3. We could also observe that the

DG scheme with k = 2 is more efficient than the one with k = 1, and the error increases

when T and b become larger.

In practice, we may need to simulate the solution with one certain stochastic path, i.e.,

M = 1. We show the results with M = 1 in Table 3 and Table 4 at T = 2π.

29

Table 2: Accuracy on (5.1) with k = 2, M = 10000, u0(x) = sin(x)

b = 0.1 b = 0.5 b = 1.0


T = 0.5

10 2.12E-03 - 2.30E-03 - 3.04E-03 -

20 2.70E-04 2.97 2.94E-04 2.97 3.97E-04 2.94

40 3.37E-05 3.00 3.67E-05 3.00 4.93E-05 3.01

80 4.22E-06 3.00 4.57E-06 3.01 5.97E-06 3.05

160 5.27E-07 3.00 5.70E-07 3.00 7.54E-07 2.99

320 6.58E-08 3.00 7.14E-08 3.00 9.56E-08 2.98

T = 1.0

10 2.58E-03 - 4.36E-03 - 1.17E-02 -

20 3.17E-04 3.03 5.17E-04 3.08 1.42E-03 3.04

40 3.93E-05 3.01 6.41E-05 3.01 1.81E-04 2.97

80 4.92E-06 3.00 8.12E-06 2.98 2.31E-05 2.97

160 6.14E-07 3.00 1.01E-06 3.01 2.88E-06 3.00

320 7.67E-08 3.00 1.25E-07 3.01 3.50E-07 3.04

T = 2.0

10 1.15E-02 - 2.87E-02 - 1.14E-01 -

20 1.32E-03 3.12 3.16E-03 3.18 1.39E-02 3.03

40 1.61E-04 3.04 3.77E-04 3.06 1.97E-03 2.82

80 2.00E-05 3.01 4.69E-05 3.01 2.09E-04 3.24

160 2.50E-06 3.00 5.90E-06 2.99 2.55E-05 3.03

320 3.12E-07 3.01 7.36E-07 3.00 3.10E-06 3.04

According to Table 3 and Table 4, we observe that if the stochastic noise of the concerning

problem is small enough, such as the case b = 0.001, the result with single path is as good

as the classical deterministic case.

We also consider the case that initial condition is discontinuous

u0(x) =

1, if π2≤ x ≤ 3π

2,

0, if 0 ≤ x < π2

or 3π2< x ≤ 2π.

We plot the approximate solution and the true solution at T = 2π with only one realiza-

tion M = 1 to get figure 1.

In view of figure 1, we could see that the DG scheme works well for the discontinuous

case and the numerical solution approximates the true solution more accurately when k

and N become larger. Similar to the deterministic cases, there are oscillations arising near

discontinuities of the solution.

Remark 5.1. For the discontinuous cases, the L2-stability, although helpful, is not enough

to control spurious numerical oscillations near discontinuities. In practice, especially for

problems containing strong discontinuities, it is worth trying to apply nonlinear limiters to

control these oscillations, which is an interesting future work to accomplish.

30

Table 3: Accuracy on (5.1) with k = 1, M = 1, T = 2π, u0(x) = sin(x)

b = 0.001 b = 0.01 b = 0.05


10 9.76E-02 - 9.69E-02 - 9.44E-02 -

20 2.31E-02 2.08 2.34E-02 2.05 2.50E-02 1.91

40 5.56E-03 2.05 5.80E-03 2.02 6.95E-03 1.85

80 1.35E-03 2.04 1.32E-03 2.13 1.23E-03 2.50

160 3.35E-04 2.01 3.27E-04 2.02 2.95E-04 2.05

320 8.38E-05 2.00 8.60E-05 1.93 9.64E-05 1.61

640 2.09E-05 2.00 2.12E-05 2.02 2.23E-05 2.11

Table 4: Accuracy on (5.1) with k = 2, M = 1, T = 2π, u0(x) = sin(x)

b = 0.001 b = 0.01 b = 0.05


10 8.59E-03 - 8.72E-03 - 9.31E-03 -

20 1.05E-03 3.03 1.05E-03 3.06 1.00E-03 3.22

40 1.32E-04 3.00 1.31E-04 3.00 1.25E-04 3.00

80 1.65E-05 3.00 1.68E-05 2.96 1.82E-05 2.78

160 2.06E-06 3.00 2.06E-06 3.03 2.05E-06 3.15

320 2.56E-07 3.00 2.52E-07 3.03 2.33E-07 3.14

640 3.22E-08 2.99 3.28E-08 2.94 3.55E-08 2.72

5.2 Linear stochastic equation with a variable coefficient

In the following we test the accuracy of the DG scheme (3.11) for the linear equation with

a variable coefficient du+ ∂∂x

(sin(x) · u

)dt = bu dWt in Ω× [0, 2π]× (0, T ),

u(ω, x, 0) = sin(x), ω ∈ Ω, x ∈ [0, 2π].(5.2)


u(ω, x, t) = v(x, t)ebWt(ω)− 12b2t,

where v is the unique solution of the following deterministic equation vt + ∂∂x

(sin(x) · v

)= 0 in [0, 2π]× (0, T ),

v(x, 0) = sin(x), x ∈ [0, 2π].(5.3)

In Table 5 and Table 6, we show the error of the DG scheme (3.11) with 10000 realizations

for k = 1 and 1000 realizations for k = 2, respectively.

31

Figure 1: Figures on (5.1) with M = 1, b = 0.5.

By Table 5 and Table 6 we observe that the order of accuracy converges to k + 1 when

N increases. Similar to the constant-coefficient linear case, the error increases when T and b

increase. The scheme with k = 2 is more efficient than the one with k = 1. All of the results

are consistent with Theorem 3.3.

5.3 Stochastic Burgers equation

Although we cannot give the error estimates for the fully nonlinear problems with locally

Lipschitz-continuous physical flux, it is worth trying to apply the DG scheme (3.1) to some

nonlinear equation. So the next example is stochastic Burgers equation du+ ∂∂x

(12u2)dt = b dWt in Ω× [0, 2π]× (0, T ),

u(ω, x, 0) = sin(x), ω ∈ Ω, x ∈ [0, 2π].(5.4)

32

Table 5: Accuracy on (5.2) with k = 1, M = 10000

b = 0.1 b = 0.5 b = 1.0


T = 0.3

10 5.13E-02 - 5.33E-02 - 6.01E-02 -

20 1.45E-02 1.82 1.51E-02 1.82 1.69E-02 1.83

40 3.94E-03 1.88 4.08E-03 1.88 4.55E-03 1.89

80 1.04E-03 1.93 1.08E-03 1.92 1.21E-03 1.91

160 2.66E-04 1.96 2.76E-04 1.97 3.09E-04 1.97

320 6.74E-05 1.98 6.96E-05 1.99 7.75E-05 1.99

T = 0.6

10 1.06E-01 - 1.14E-01 - 1.42E-01 -

20 2.95E-02 1.84 3.17E-02 1.84 3.96E-02 1.84

40 8.22E-03 1.84 8.83E-03 1.84 1.10E-02 1.85

80 2.20E-03 1.90 2.36E-03 1.90 2.95E-03 1.90

160 5.71E-04 1.94 6.17E-04 1.94 7.79E-04 1.92

320 1.46E-04 1.97 1.57E-04 1.98 1.98E-04 1.98

T = 0.9

10 2.36E-01 - 2.64E-01 - 3.65E-01 -

20 6.00E-02 1.98 6.70E-02 1.98 9.31E-02 1.97

40 1.70E-02 1.82 1.90E-02 1.82 2.61E-02 1.84

80 4.63E-03 1.88 5.13E-03 1.89 6.96E-03 1.90

160 1.22E-03 1.93 1.36E-03 1.92 1.86E-03 1.90

320 3.13E-04 1.96 3.48E-04 1.97 4.84E-04 1.95


u(ω, x, t) = v

(x− b

∫ t

0

Ws ds, t

)+ bWt(ω),

where v is the solution of the following deterministic equation vt + ∂∂x

(12v2)

= 0 in [0, 2π]× (0, T ),

v(x, 0) = sin(x), x ∈ [0, 2π],(5.5)

and the random variable∫ t

0Ws ds could be computed exactly by (4.7).

We use the simple Lax-Friedrichs flux

f(u−, u+

)=

1

4

(u−)2

+(u+)2− 1

2α(u+ − u−

),

where

α = maxj

∣∣∣u−j+ 1

2

∣∣∣ , ∣∣∣u+j+ 1

2

∣∣∣ .In Table 7 and Table 8, we show the errors of the DG scheme (3.1) with 100 realizations.

33


b = 0.1 b = 0.5 b = 1.0


T = 0.3

10 4.74E-03 - 4.90E-03 - 5.50E-03 -

20 6.54E-04 2.86 6.76E-04 2.86 7.48E-04 2.88

40 8.82E-05 2.89 9.15E-05 2.89 1.01E-04 2.89

80 1.17E-05 2.91 1.21E-05 2.92 1.34E-05 2.92

160 1.54E-06 2.93 1.61E-06 2.91 1.78E-06 2.91

320 2.02E-07 2.93 2.09E-07 2.95 2.32E-07 2.94

T = 0.6

10 1.21E-02 - 1.30E-02 - 1.62E-02 -

20 2.08E-03 2.54 2.24E-03 2.54 2.80E-03 2.53

40 2.86E-04 2.87 3.09E-04 2.86 3.92E-04 2.84

80 3.85E-05 2.89 4.15E-05 2.89 5.25E-05 2.90

160 5.10E-06 2.92 5.49E-06 2.92 6.66E-06 2.98

320 6.63E-07 2.94 7.04E-07 2.97 8.43E-07 2.98

T = 0.9

10 1.78E-02 - 1.97E-02 - 2.76E-02 -

20 6.07E-03 1.55 6.64E-03 1.57 9.01E-03 1.61

40 8.47E-04 2.84 9.56E-04 2.80 1.35E-03 2.73

80 1.15E-04 2.88 1.26E-04 2.92 1.68E-04 3.01

160 1.54E-05 2.90 1.73E-05 2.87 2.36E-05 2.83

320 2.01E-06 2.94 2.21E-06 2.97 2.92E-06 3.01

Notice that the solution of (5.5) has an infinite slope - the wave “breaks” and a shock

forms at

Tb =−1

min v′0(x)= 1 ,

see [21].

From Table 7 and Table 8, we observe that the order of accuracy converges to k+1 when

N increases for the case that T < Tb. The scheme with k = 2 is more efficient than the one

with k = 1.

Unlike the diffusion effect of the stochastic terms on the solutions of (5.1) and (5.2),

here the stochastic term only has the drift effect on the solution of (5.4) since the stochastic

perturbation in (5.4) is additive. Thus the value of b has little influence on the error and

M = 100 is good enough to approximate the mathematical expectation.

When T increases, the scheme converges slowly and becomes more inefficient. When

T > Tb, the DG scheme lose its order of accuracy. To see the behaviour of the approximate

solution with T > Tb, we plot the approximate solution and the true solution at T = 1.5

with only one realization M = 1 to get figure 2.

From figure 2, we see that the DG schemes works well and the numerical solution ap-

proximates the true solution more accurately when k and N increase.

34


b = 0.1 b = 1.0 b = 2.0


T = 0.1

10 3.18E-02 - 3.19E-02 - 3.19E-02 -

20 9.35E-03 1.77 9.30E-03 1.78 9.17E-03 1.80

40 2.59E-03 1.85 2.58E-03 1.85 2.52E-03 1.86

80 6.70E-04 1.95 6.68E-04 1.95 6.56E-04 1.94

160 1.69E-04 1.99 1.68E-04 1.99 1.65E-04 1.99

320 4.23E-05 2.00 4.22E-05 1.99 4.15E-05 1.99

T = 0.5

10 5.66E-02 - 5.47E-02 - 5.51E-02 -

20 1.52E-02 1.90 1.50E-02 1.87 1.52E-02 1.86

40 4.08E-03 1.90 3.99E-03 1.91 4.09E-03 1.89

80 1.07E-03 1.93 1.04E-03 1.93 1.05E-03 1.96

160 2.75E-04 1.96 2.69E-04 1.96 2.71E-04 1.96

320 6.97E-05 1.98 6.85E-05 1.97 6.85E-05 1.98

T = 0.9

10 1.75E-01 - 1.81E-01 - 1.95E-01 -

20 7.74E-02 1.18 7.74E-02 1.22 8.72E-02 1.16

40 2.69E-02 1.53 2.98E-02 1.38 3.54E-02 1.30

80 8.51E-03 1.66 9.64E-03 1.63 1.17E-02 1.60

160 2.47E-03 1.78 2.85E-03 1.76 3.43E-03 1.77

320 6.90E-04 1.84 7.58E-04 1.91 8.86E-04 1.95

T = 1.5

10 4.30E-01 - 4.36E-01 - 4.45E-01 -

20 3.01E-01 0.52 3.00E-01 0.54 3.13E-01 0.51

40 2.03E-01 0.57 2.03E-01 0.57 2.06E-01 0.60

80 1.31E-01 0.63 1.29E-01 0.65 1.34E-01 0.62

160 8.43E-02 0.63 8.17E-02 0.66 8.18E-02 0.71

320 7.48E-02 0.17 6.86E-02 0.25 6.49E-02 0.33

6 Concluding remarks

In this article, we present semi-discrete DG schemes for fully nonlinear stochastic equations

and semilinear variable-coefficient stochastic equations. We obtain the L2 stability results of

the schemes. We prove the optimal error estimates of order O(hk+1) for semilinear stochastic

conservation laws with variable coefficients. We also establish an explicit derivative-free

second order time discretization scheme and perform several numerical experiments on some

model problems to confirm the analytical results. It is more challenging to investigate error

estimates for fully nonlinear stochastic equations and apply the DG type schemes to SPDEs

with high-order spatial derivatives, which will be carried out in the future.

35


b = 0.1 b = 1.0 b = 2.0


T = 0.1

10 1.67E-03 - 1.68E-03 - 1.72E-03 -

20 2.38E-04 2.81 2.41E-04 2.80 2.52E-04 2.77

40 3.44E-05 2.79 3.47E-05 2.80 3.55E-05 2.83

80 4.39E-06 2.97 4.47E-06 2.96 4.61E-06 2.95

160 5.53E-07 2.99 5.70E-07 2.97 5.85E-07 2.98

320 6.97E-08 2.99 7.17E-08 2.99 7.29E-08 3.00

T = 0.5

10 3.53E-03 - 6.59E-03 - 7.79E-03 -

20 8.64E-04 2.03 1.01E-03 2.71 1.08E-03 2.85

40 1.24E-04 2.80 1.41E-04 2.84 1.49E-04 2.86

80 1.67E-05 2.89 1.88E-05 2.91 1.96E-05 2.92

160 2.24E-06 2.90 2.46E-06 2.93 2.52E-06 2.96

320 2.95E-07 2.93 3.16E-07 2.96 3.22E-07 2.97

T = 0.9

10 6.80E-02 - 8.78E-02 - 9.86E-02 -

20 2.29E-02 1.57 3.18E-02 1.46 3.51E-02 1.49

40 7.94E-03 1.53 9.17E-03 1.79 1.12E-02 1.65

80 1.74E-03 2.19 2.00E-03 2.20 2.40E-03 2.23

160 2.62E-04 2.73 3.29E-04 2.60 3.88E-04 2.63

320 3.78E-05 2.79 4.70E-05 2.81 5.01E-05 2.95

T = 1.5

10 4.02E-01 - 3.58E-01 - 3.58E-01 -

20 2.88E-01 0.48 2.47E-01 0.54 2.41E-01 0.57

40 1.91E-01 0.59 1.66E-01 0.57 1.64E-01 0.55

80 1.63E-01 0.23 1.08E-01 0.62 1.05E-01 0.64

160 1.41E-01 0.20 8.60E-02 0.33 7.69E-02 0.45

320 1.43E-01 -0.02 9.04E-02 -0.07 9.18E-02 -0.26

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