A REVIEW

NUiVlERICAL METHODS FOR NONLINEAR PDES

~

517

';/;,

~I o

17­ ) of

unit

80,.

)) + )=

dimensional 5envalues of

haracteriza­

such that = G,6" can ;ure 3.3. 1dent prob­ tesian grid , e-difference 5

n ')w k jk '

jes the dif­ g from one :onvection­ a

~xample,

FIGURE 3.4. The time evolution of vorticity in two-dimensional inviscid Euler equations (2.8b) using a central difference scheme [137] computed at t = 4,6 and t = 10.

Figure 3.4 demonstrates a finite-difference computation of the vorticity equation (2.8b). Often, one is interested in discretizing only the spatial variables. For example, a finite-difference discretization of the two-dimensional chemotaxis problem (2.9) reads (3.8)

:tWj = -K:(D+x(Hl.iD-xCj ) +D+y(WjD_yCj ))

+ (D+xD_x+D+yD_y)l1;j.

Here, W j == W(j,,]2) (t) are the approximate densities at the gridpoints Xj E D,6,. The missing boundary values {Wj(t), Xj E 8D,6,} are recovered from the Neumann-typ e boundary conditions DnWj(t) = O. The approximate concentrations Cj == C j ¡,j2(t) are obtained as a finite-difference solution of the Poisson equation (2.9b), based on the standard five-point stencil interior scheme : boundary conditions :

(D+xD-x + D+yD_y)Cj(t) DnCj

-Wj(t) , Xj E D,6" O, Xj E 8D,6, .

In this fashion, one ends up with a semidiscrete approximation (3.8), called the method of lines, which amounts to a nonlinear system of ordinary differential equa­ tions (ODEs) for the unknowns {Wj (t)} . The solution of such semidiscrete systems is obtained by standard ODE solvers [82 , 96, 97, 28]. The finite-difference schemes (3.2)-(3.8) are typical examples of finite-difference approximations of nonlinear PDEs. The general recipe for such schemes can be ex­ pressed in terms of divided difference operators of order j, oi,6, = D~X l D!:X2 ... D!:Xd' which are supported on the computational grid D,6, . A finite-difference approximation of the PDE (2.1) is obtained by replacing the nonlinear relations between partial derivatives of w in (2.1) with similar relations between divided differences of the gridfunction, W,6"

(3.9)

A(D~ W j ) D~-lWj, . . . , D,6, W j , W j , Xj) = G j ,

Xj = (X.i,' X.i2'···' X,id) E D,6,

e lR~.

Here, W,6, = {Wj } is the computed gridj'unction and G j are discrete approxima­ tions of the source term, G j ~ g(Xj). We can distinguish between two main classes of finite-difference methods: