## Abstract

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 guarantees a smooth solution at all later times, such as the barotropic vorticity equation (3.123), any difficulties associated with poor numerical resolution can be avoided by using a sufficiently fine computational mesh. But if the governing equations allow an initially smooth field to develop shocks or discontinuities, as is the case with Burgers’s equation (3.113), 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 Chapter 2 to guarantee that the numerical solution converges to the correct solution as the spatial grid interval and the time step approach zero.

## Keywords

Weak Solution Rarefaction Wave Advection Equation Courant Number Flux Limiter## Preview

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## References

- 1.The jump is “downward” in the sense that the fluid level drops during the passage of the discontinuity.Google Scholar
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*Negative definite*schemes may be similarly defined as any method that never generates positive values from nonpositive initial data. Any positive definite scheme can be trivially converted to a negative definite method.Google Scholar