Conservation Laws and Finite-Volume Methods

As demonstrated in the preceding chapters, the errors in most numerical solutions increase dramatically as the physical scale of the simulated disturbance approaches the minimum scale resolvable on the numerical mesh. When solving equations for which smooth initial data guarantee a smooth solution at all later times, such as the barotropic vorticity equation (4.112), one can avoid any difficulties associated with poor numerical resolution by using a sufficiently fine computationalmesh. But if the governing equations allow an initially smooth field to develop shocks or discontinuities, as is the case with Burgers’s equation (4.102), there is no hope of maintaining adequate numerical resolution throughout the simulation, and special numerical techniques must be used to control the development of overshoots and undershoots in the vicinity of the shock. Numerical approximations to equations with discontinuous solutions must also satisfy additional conditions beyond the stability and consistency requirements discussed in Chap. 3 to guarantee that the numerical solution converges to the correct solution as the spatial grid interval and the time step approach zero.