7 Numerical Stability

In the previous sections we have discussed continuum initial and IBVPs. In this section we start with the study of the discretization of such problems. In the same way that a PDE can have a unique solution yet be ill posed27, a numerical scheme can be consistent yet not convergent due to the unbounded growth of small perturbations as resolution is increased. The definition of numerical stability is the discrete version of well-posedness. One wants to ensure that small initial perturbations in the numerical solution, which naturally appear due to discretization errors and finite precision, remain bounded for all resolutions at any given time t > 0. Due to the classical Lax–Richtmyer theorem [276], this property, combined with consistency of the scheme, is equivalent in the linear case to convergence of the numerical solution, and the latter approaches the continuum one as resolution is increased (at least within exact arithmetic). Convergence of a scheme is in general difficult to prove directly, especially because the exact solution is in general not known. Instead, one shows stability.

The different definitions of numerical stability follow those of well-posedness, with the L2 norm in space replaced by a discrete one, which is usually motivated by the spatial approximation. For example, discrete norms under which the summation by parts property holds are natural in the context of some finite difference approximations and collocation spectral methods (see Sections 8 and 9).

We start with a general discussion of some aspects of stability, and explicit analyses of simple, low-order schemes for test models. There follows a discussion of different variations of the von Neumann condition, including an eigenvalue version, which can be used to analyze in practice necessary conditions for IBVPs. Next, we discuss a rather general stability approach for the method of lines, the notion of time-stability, Runge–Kutta methods, and we close the section with some references to other approaches not covered here, as well as some discussion in the context of numerical relativity.

 7.1 Definitions and examples
 7.2 The von Neumann condition
  7.2.1 The periodic, scalar case
  7.2.2 The general, linear, time-independent case
 7.3 The method of lines
  7.3.1 Semi-discrete stability
  7.3.2 Fully-discrete stability
 7.4 Strict or time-stability
 7.5 Runge–Kutta methods
  7.5.1 Embedded methods
 7.6 Remarks

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