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8.1 The Finite Difference Approximation

The finite difference equation, based on second order in time and second order in space approximations, is

Equation 47 is the discrete approximation to the wave equation shown in Equation 1. The choice of the right hand side (RHS) time value as n describes an explicit time marching scheme where the n+1 is a function of n and n-1:

If the RHS was chosen to have an n+1 time value, the resulting difference equation would have been implicit. The implicit form requires solving a system of equations at each time step and has a significant increase in computational complexity. The explicit form Equation 48 is used here with good results, and in a later section the stability of the explicit method is considered.