Numerical Dissipation

Abstract

Up to this point, we have emphasized the second-order centered-difference approximations to the spatial derivatives in our model equations. We have seen that a centered approximation to a first derivative is nondissipative, i.e. the eigenvalues of the associated circulant matrix (with periodic boundary conditions) are pure imaginary. In processes governed by nonlinear equations, such as the Euler and Navier—Stokes equations, there can be a continual production of high-frequency components of the solution, leading, for example, to the formation of shock waves. In a real physical problem, the production of high frequencies is eventually limited by viscosity. However, when we solve the Euler equations numerically, we have neglected viscous effects. Thus the numerical approximation must contain some inherent dissipation to limit the production of high-frequency modes. Although numerical approximations to the Navier—Stokes equations contain dissipation through the viscous terms, this can be insufficient, especially at high Reynolds numbers, due to the limited grid resolution which is practical. Therefore, unless the relevant length scales are resolved, some form of added numerical dissipation is required in the numerical solution of the Navier—Stokes equations as well. Since the addition of numerical dissipation is tantamount to intentionally introducing nonphysical behavior, it must be carefully controlled such that the error introduced is not excessive. In this chapter, we discuss some different ways of adding numerical dissipation to the spatial derivatives in the linear convection equation and hyperbolic systems of PDE’s.