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146 5. Numerical Methods for Model Hyperbolic Equations

T
2 1
X- 1 = [L'L ] ' 1 1 " "l u + c (E5.1.6)
Ai A 2 1 u — c
where
2
c = y/u +f3
The right eigenvector matrix X of A is
1 u — c —u — c
1 _ - f
X = (X' (E5.1.7)
2c [ - 1 1
and
[Ai u + c
l
X~ AX = A = (E5.1.8)
A 2 u — c
In this chapter we address the numerical solution of the scalar equation
(5.1.1) or the vector equation (5.1.4) with explicit methods in Sections 5.2 and
5.3, implicit methods in Section 5.4 and upwind methods in Section 5.5. Finite
volume methods are discussed in Section 5.6 and the convergence and stability
of the finite-difference methods in Section 5.7.

5.2 Explicit Methods: Two-Step Lax-Wendroff Method

There are a number of explicit methods developed for hyperbolic equations and
an excellent and detailed description of them is given by Hirsch [1]. Among
these methods, the two-step Lax-Wendroff is the most important of them, due
to its uniqueness for linear equations and its important role as the guideline for
the development of many schemes to improve some of its deficiencies and its
extension to nonlinear equations.
The two-step Lax-Wendroff method is second-order accurate and consists
of two steps. (A one-step method employs only data at t = t n to obtain an ap-
proximation to u(x, t) at t = t n+1 .) In step 1, the values of u(x, t) are computed
n
/ ,
at t = £ n + 1 2 by discretizing the flux term by central differences at (xi+i/2,t )
2
and the time term by forward difference at (^+i/2^ n + 1 ^ ) 5 that is,
n+l/2 „
V l / 2 " ^ + 1 / 2 c < + l 4
= 0 (5.2.1a)
At/2 Ax
or n+l/2
Vhl/2 u i+l/2 M +i (5.2.1b)
where
1
< + i / 2 = zW + Ui+i)
The parameter a is the Courant number, also called the CFL (Courant-
Friedrichs-Lewy) number
At
a = c (5.2.2)
Ax
discussed in more detail in Section 5.7.