Explicit Finite-Difference Methods

Abstract

If you restrict yourself to using the same grid points as those in section 4.2, a general explicit method can be written as

$$\frac{{c_j^{n + 1} - \frac{1}{2}\alpha \left( {c_{j + 1}^n + c_{j - 1}^n} \right) - \left( {1 - \alpha } \right)c_j^n}}{{\Delta t}} + u\frac{{c_{j + 1}^n - c_{j - 1}^n}}{{2\Delta x}} = 0$$

((5.1))

where α is a free parameter that can be manipulated for stability and accuracy. An exercise for this chapter shows that this is indeed a general method for approximation of the simple-wave equation without a decay term. If α=1, you get the method of Lax; therefore the general case can be called a modified Lax method. The difference equation used in section 4.2 is a special case α=0.