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.
KeywordsWeak Solution Rarefaction Wave Advection Equation Courant Number Flux Limiter
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- 1.The jump is “downward” in the sense that the fluid level drops during the passage of the discontinuity.Google Scholar
- 2.Any differentiable function is Lipschitz continuous.Google Scholar
- 4.Equation 5.25 states that the tracer concentration (typically expressed as a dimensionless ratio, such as grams per kilogram or parts per billion) is conserved following the motion of each fluid parcel. In contrast, (5.23) states that the local rate of change of the mass of the tracer at a fixed point in space is determined by the divergence of the tracer mass-flux at that point.Google Scholar
- 5.The general case, in which the phase speed is either positive or negative, is discussed in Section 5.5.3.Google Scholar
- 7.See LeVeque (1992) for a discussion of approximate techniques for the solution of Riemann problems involving nonlinear systems of conservation laws.Google Scholar
- 8.The values of the prognostic variables are required at every time step during the integration of systems of equations in which a chemical or physical process (such as cloud condensation and precipitation) is parametrized as a function of the prognostic variables.Google Scholar
- 9.See Fig. 3.6 for an illustration of the same staggering scheme in a different context.Google Scholar
- 10.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