py4066 partial differential equations · py4066 partial differential equations ii recommended texts...
TRANSCRIPT
![Page 1: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/1.jpg)
PY4066 Partial Differential Equations IIRecommended texts
• Partial differential equations for scientists and engineersS.J. Farlow, Dover (1993) S-LEN 515.353 M23
• Partial differential equations for scientists and engineersG. Stephenson, Imperial College Press (1996) 515.353 N8
• Elements of soliton theoryG.L. Lamb, Wiley (1980)515.353 M05
![Page 2: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/2.jpg)
Course Outline
• 1 Computer modelling of fluids– Fluid types, transport, conservation laws, wave equations
• 2 Nonlinear wave equations– Solitary waves and solitons, KdV equation, NLS equation
• 3 Solution of PDE’s using integral transforms– Fourier, sine, cosine and Laplace transforms
• 4 Solution of PDE’s using variational principles– Euler-Lagrange equations, method of Ritz, finite element method
![Page 3: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/3.jpg)
1 Computer Modelling of Fluids1.1 Introduction
• Fluid types– Quantum fluids, classical fluids
• Modes of transport– Densities (number, charge, energy, …)– Currents (flows of mass, charge, energy)– Continuity equation relates current and density– Advection, convection, diffusion
• Conservation laws– Mass, momentum, energy, …
• Wave equations
![Page 4: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/4.jpg)
•Fick’s law (diffusive transport)t),( D- t),( rrj ρ∇=
1.2 Diffusive, Convective and Advective Transport
•Diffusion equation
t),f( t),( D- t
t),( 2 rrr =∇∂
∂ ρρ
•Substitute Fick’s law into continuity equation to obtain diffusion equation
•Continuity equation with sources
t),f( t
t),( t),(. rrrj =∂
∂+∇ ρ
![Page 5: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/5.jpg)
Diffusive, Convective and Advective Transport
• Advection (advective transport)
–Molecular motion irrelevantt),v(t),( t),( rrrj ρ=
• Advection-diffusion equation
t),f( t)],t)v(,([ . t),( D- t
t),( 2 rrrrr =∇+∇∂
∂ ρρρ
• Add divergence of advective flux term (when both types of transport are present) to obtain advection-diffusion
A solute (e.g. pollutant) added to a flowing river is subject to advective (flow with river) and diffusive transport (diffusion in water).
![Page 6: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/6.jpg)
Diffusive, Convective and Advective Transport
• Convection occurs when an element of fluid is subjected to an external force owing to an intrinsic property of the fluid element
• For example, if a fluid is heated from below in a gravitational field, a layer of cooler and denser fluid overlies a warmer and less dense fluid. This instability relaxes either bydiffusion of heat (without any transport of the fluid) or by convection, in which case heat is transported along with the fluid
• The Lorenz equations* are a good example
of convection in a simple system *To go to this webpage, click on the link when in ‘show’ mode (F5 k )
![Page 7: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/7.jpg)
1.3 Wave motion in channels
•We will derive coupled, partial differential equations representing conservation of mass and conservation of momentum for water in a channel of width, b.
•The purpose of this is to see where pde’s come from
•Approximate forms for the coupled, nonlinear pde’s which represent the conservation laws can be chosen which lead to wave equations.
![Page 8: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/8.jpg)
Wave motion in channelsConservation laws
•Momentum conservation (Newton’s laws)
•Mass conservation (continuity equation)
•Energy conservation (equations of state)
•These generally result in sets of coupled, nonlinear pde’s
•Under particular circumstances these may result in wave motion, turbulence, etc.
![Page 9: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/9.jpg)
Wave motion in channelsConservation of mass
• Continuity equation is applied to water in the channel with width, b, height, h(x,t), thickness, ∆x, moving parallel to x direction at velocity u(x,t). Water density is ρ
![Page 10: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/10.jpg)
Wave motion in channelsConservation of mass
• Mass of slice = volume . density = b ∆x h ρSubstitute areal density into continuity equation
( ) ( )
( )
equation mass of onConservati 0 xuh
xhu
th
0 t)u(x, t)h(x, x
t)h(x,t
0 t)u(x, ρ t)h(x,∆x bx
ρ t)h(x,∆x bt
=∂∂
+∂∂
+∂∂
=∂∂
+∂∂
=∂∂
+∂∂
![Page 11: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/11.jpg)
Wave motion in channelsConservation of momentum
• Consider a slice where the height of the slice varies within the slice. There is a pressure difference across the slice which tends to accelerate the slice laterally• Compute the pressure difference across the slice
• Recognize that this corresponds to a force, F(x,t)
• Substitute this force into Newton’s 3rd Law (conservation of momentum)
t)F(x, momentum)(dtd =
t)h(x,
x1x 2x
![Page 12: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/12.jpg)
Wave motion in channelsConservation of momentum
dx t)u(x, t)h(x, b ρ
(t)x
(t)x
2
1
∫
• Instantaneous momentum of the slice
x2(t) – x1(t) = ∆x
• Differentiate wrt time using Leibniz’ rule
t),u(x t),(t)h(xx - t),u(x t),h(x (t)x
dx )t)u(x, t)(h(x,t
dx t)u(x, t)h(x, dtd
111
.
222
.
(t)x
(t)x
(t)x
(t)x
2
1
2
1
+∂∂
= ∫∫
![Page 13: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/13.jpg)
Wave motion in channelsConservation of momentum
2
2
x)O( x
t),h(x ) x- (x t),h(x t),h(x
x)O( x
t),u(x ) x- (x t),u(x t),u(x (t)x
t),u(x (t)x
11212
112122
.
11
.
∆+∂
∂+=
∆+∂
∂+==
=
steps) of detailsfor sheet Tutorial (see
x)O( x xuu
tuh b dx t)u(x, t)h(x, b
dtd
for the find werelations thesengsubstituti
2
(t)x
(t)x
2
1
∆+∆
∂∂
+∂∂
=∫ ρρ
momentum of change of rate
![Page 14: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/14.jpg)
Wave motion in channelsConservation of momentum
•Water pressure at depth, h – z, measured from the water surface is ρg (h – z)
•Pressure force averaged over a slice at (x1, t) is
•Force due to gradient of water height
( ) b dzz- t),h(xρg t),F(x
t),h(x
011
1
∫=
t)h(x,
x1x 2x
![Page 15: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/15.jpg)
Wave motion in channelsConservation of momentum
• Difference in pressure force across a slice, of width ∆x, by Taylor series expansion is
( )
2
2t),h(x
0
1
2t),h(x
0112
x)O(x bxhh ρg
x)O( ∆x b dzx
t),h(xρg
x)O( ∆x b dzz- t),h(xdxdρg t),F(x - t),F(x
1
1
∆∆∂∂
=
∆+
∂
∂=
∆+
=
∫
∫
![Page 16: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/16.jpg)
Wave motion in channelsConservation of momentum
• Net force on the slice in the positive x direction is
2x)O( ∆x xhgρbh - ∆+∂∂
• Rate of change of momentum of the slice
2x)O( ∆x xuu
tuρbh ∆+
∂∂
+∂∂
• Equate these and divide by common factors
0 xhg
xuu
tu =
∂∂
+∂∂
+∂∂
![Page 17: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/17.jpg)
Wave motion in channelsResulting equations
Mass ofon Conservati 0 xuh
xhu
th
Momentum ofon Conservati 0 xhg
xuu
tu
=∂∂
+∂∂
+∂∂
=∂∂
+∂∂
+∂∂
A pair of coupled, nonlinear partial differential equations
![Page 18: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/18.jpg)
Wave motion in channelsDerivation of wave equations
• Let the water height h be where h is the mean depthand h is the local wave height
• In the limit:– water is deep compared to the wave height – these equations reduce to the usual, linear wave equation
• In the limits: and– wavelength is long compared to water depth ( )– a nonlinear wave equation with solitary wave solutions
can be derived (see Lamb pp 169 ff for details)
η+=hh
h<<η
h>>λ h<<ηh>>λ
![Page 19: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/19.jpg)
Wave motion in channelsDerivation of wave equations
• If we can neglect and in conservation equations
h<<ηxuu∂∂
xu∂∂η
• For these becomeh<<η
Mass of onConservati
Momentum of onConservati
0 xuh
t
0 x
g tu
=∂∂
+∂∂
=∂∂
+∂∂
η
η
![Page 20: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/20.jpg)
Wave motion in channelsDerivation of wave equations
( )0
th
1x
0 tx
uh t
0 xuh
tt
0 x
g tx
u 0 x
g tu
x
2
2
22
2
2
2
2
2
22
=∂∂
−∂∂
=∂∂
∂+
∂∂
⇒=
∂∂
+∂∂
∂∂
=∂∂
+∂∂
∂⇒=
∂∂
+∂∂
∂∂
ηη
ηη
ηη
g
• Solutions are waves with velocity hg
![Page 21: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/21.jpg)
1.4 Solution of coupled nonlinear pde’s
• Apply leapfrog method for solving pde’s in more than one variable
• Apply forward and/or central finite difference schemes
![Page 22: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/22.jpg)
Solution of coupled nonlinear pde’sLeapfrog methods for coupled equations
• Solve for u and h in alternate timesteps
![Page 23: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/23.jpg)
Solution of coupled nonlinear pde’sCentral difference leapfrog scheme
xg
tu
∂∂
+∂∂ η
onConservati Momentum
0 x2
hhg
t2
uu i1-j i
1j1i
j 1ij =
∆+
∆
−+−−+
( ) hhxtg - uu i
1-j i1j 1i
j 1ij −+
−+∆∆
=
![Page 24: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/24.jpg)
Solution of coupled nonlinear pde’sCentral difference leapfrog scheme
xuh
t ∂∂
+∂∂η
onConservati Mass
0 x2
uuh
t2
hh i1-j i
1j1i
j 1ij _
=∆
+∆
−+−−+
( ) uuxth - hh i
1-j i1j 1i
j 1ij
_−+
−+∆∆
=
![Page 25: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/25.jpg)
Solution of coupled nonlinear pde’sForward/Central difference leapfrog scheme
xg
tu
∂∂
+∂∂ η
onConservati Momentum
0 x2
hhg
t
uu 1i1-j 1i
1jij 1i
j =∆
+∆
+−++−+
( ) hhx2tg - uu 1i
1-j 1i1j i
j 1ij
+−++
+∆∆
=
![Page 26: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/26.jpg)
Solution of coupled nonlinear pde’sForward/Central difference leapfrog scheme
xuh
t ∂∂
+∂∂η
onConservati Mass
0 x2
uuh
t
hh i1-j i
1jij 1i
j _=
∆+
∆
−+−+
( )i1-j i
1j ij 1i
j uux2th - hh
_−+
+∆∆
=
![Page 27: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/27.jpg)
Solution of coupled nonlinear pde’sForward/central difference leapfrog scheme
Momentum ofon Conservati 0 xhg
xuu
tu =
∂∂
+∂∂
+∂∂
0 x2
hhg
x2
uuu
t
uu 1i1-j 1i
1ji
1-j i1ji
j
ij 1i
j =∆
+∆
+∆
+−++−+−+
( ) ( )[ ] hhg uuux2t - uu 1i
1-j 1i1j
i1-j i
1jij i
j 1ij
+−++−+
+ +∆∆
=
![Page 28: PY4066 Partial Differential Equations · PY4066 Partial Differential Equations II Recommended texts • Partial differential equations for scientists and engineers S.J. Farlow, Dover](https://reader030.vdocuments.site/reader030/viewer/2022012320/5b7d1b1f7f8b9a717e8be0f3/html5/thumbnails/28.jpg)
Solution of coupled nonlinear pde’sForward/central difference leapfrog scheme
Mass ofon Conservati 0 xuh
xhu
th
=∂∂
+∂∂
+∂∂
0 x2
uuh
x2
hhu
t
hh i1-j i
1jij
i1-j i
1jij
ij 1i
j =∆
+∆
+∆
−+−+−+
( ) ( )[ ]i1-j i
1jij
i1-j i
1jij i
j 1ij uuhhhu
x2t - hh −+−+
+ +∆∆
=