椭圆形偏微分方程的数值方法

2. Poisson’s Equation
∂2u ∂2u
+ = f (x, y)
(6)
∂x2 ∂y2
often written as ▽2u = f (x, y). Physical examples of Poisson’s equation is steady state heat conduction with some internal heat source and the solution of Gauss’s law for a conservative field.
To fully determine the solution of equation (1) we will need one boundary condition on
the whole of the boundary, S. There are three typical boundary conditions as follows:
u(x + h) = u(x) + hu′(x) + h2 u′′(x) + h3 u′′′(x) + h4 u′′′′(x) + . . .,
2!
3!
4!
and,
PDEs: Elliptic Equations
4
u(x − h) = u(x) + hu′(x) + h2 u′′(x) − h3 u′′′(x) + h4 u′′′′(x) − . . .,
u r - 1,s
u r,s+1 u r,s
u r+1,s
u r,s-1
Figure 2: Finite Difference Stencil at Point (r, s).
If we take the simple case where δx = δy (square grid) then equation (9) becomes,
u = g1(x, y) − Dirichlet condition
(2)
∂u
∂n = g2(x, y) − Neumann condition
(3)
∂u
α(x, y)u + β(x, y) = γ(x, y) − mixed condition
(4)
∂n
∂ where represents differentiation along the outward normal to R.
∂2u ∂2u ∂2u
a +b
+ c = f,
(1)
∂x2 ∂x∂y ∂y2
∂u
∂u
where a, b, c are functions of x and y and f is a function of x, y, u, and .
∂x
∂y
Equation (1) will be elliptic if b2 − 4ac < 0.
Finite Difference Methods for Elliptic Partial Differential Equations∗
Aims
• To introduce finite-difference discretisations of linear elliptic PDEs on regular grids.
There are thus (n − 1) × (m − 1) internal node points where the solution is to be calculated.
2.2 PDE Discretisation
The next step is now to discretise the PDE (5). Let ur,s = u(rδx, sδy) and consider the approximation of equation (5) at the point (rδx, sδy). We use the following Taylor’s series expansions:
d2u which is the Central Difference approximation to .
dx2 Thus, in two dimensions (where partial derivatives replace full derivatives) we may write
∂2u
= u(xr + δx, ys) − 2u(xr, ys) + u(xr − δx, ys) + O(δx2),
3 Derivative boundary conditions
11
4 More accurate formulae to solve Laplace’s equation
14
4.1 A 9 Point Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Domain Discretisation
The first step is to discretise the domain R. This is done by covering R by a rectangular grid of size δx i the x direction and δy in the y direction where
Contents
1 Introduction
2
1.1 Example’s of Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . 2
Hale Waihona Puke 2 Finite Difference Approximation
3
2.1 Domain Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
PDEs: Elliptic Equations
5
Equation (9) links the value of u at the point (xr, ys) with its four immediate neighbours as depicted in Figure 2. Since 5 points are involved we say this scheme has a 5 Point Stencil.
• To show how iterative techniques may be used to solve the resulting discrete equations.
• To show how increasing the stencil size may be used to increase accuracy.
2.2 PDE Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 The Solution of Finite Difference Equations . . . . . . . . . . . . . . . 7
2!
3!
4!
so that,
So now,
u(x + h) + u(x − h) = 2u(x) + 2h2 u′′(x) + 2h4 u′′′′(x) + . . ..
2!
4!
giving
u′′(x)
=
u(x
+
h)
−
2u(x) + h2
u(x − h)
−
h2 u′′′′(x) 12
+...,
u′′(x) = u(x + h) − 2u(x) + u(x − h) + O(h2), h2
1
PDEs: Elliptic Equations
2
1 Introduction
Suppose that R is some general closed region in the (x, y) plane, with boundary S. We consider numerical methods for the solution of the equation
4.2 Alternative 9 Point Laplacian . . . . . . . . . . . . . . . . . . . . . . . 16
5 Conclusions
17
References
17
∗Adapted from notes by Robert Bennell and Shaun Forth
δy2
or,
∂2u = ur,s+1 − 2ur,s + ur,s−1 + O(δy2).
(8)
∂y2 r,s
δy2
Thus equation (5) may be replaced by the finite difference equation,
ur+1,s − 2ur,s + ur−1,s + ur,s+1 − 2ur,s + ur,s−1 = 0.
∂n
1.1 Example’s of Elliptic Equations
The best known elliptic PDEs are 1. Laplace’s Equation
∂2u ∂2u
+ =0
(5)