Stability criteria for hybrid difference methods
Introduction
Hybrid numerical methods are becoming increasingly popular for problems where different physics are important in different parts of the domain. The hybridization can take many different forms, some examples of which include: the coupling of different codes to solve different underlying PDEs in different regions (say, the compressible and incompressible Navier–Stokes equations); the use of grids with different topologies in different regions (say, structured and unstructured grids); and the use of different numerical schemes in different regions within the same code and PDE. The latter situation arises in the computation of flows with shock waves, density interfaces, and turbulence, where one might use a computationally efficient and accurate linear scheme in the turbulence regions and a more robust nonlinear shock-capturing scheme around the discontinuities. These hybrid methods are typically implemented by using one scheme for and the other for , say, where the interface location may depend on the instantaneous solution.
Ideally, a hybrid method combines the strengths of the individual methods being used, but accuracy and stability at the interface between the schemes is an issue that must be resolved. Even when both schemes are stable individually, there is no guarantee that the coupled method will be stable. One common class of hybrid methods couples an ENO or WENO scheme [1] with a centered or upwinded linear scheme. Adams and Shariff [2] used the combination ENO/upwinded Padé and noticed small oscillations (noise) around the interface. Later, Ren et al. [3] argued that the sharp transition from one scheme to the other is responsible for the noise, and proposed an approach where the schemes are smoothly blended into each other over several grid points. Hill and Pullin [4] argued that the noise around the interface can be minimized by forcing the optimal WENO stencil to equal that of the linear scheme, thus avoiding or minimizing the change across the interface.
In this paper, we analyze the stability characteristics of a general hybrid method using a linear model problem. The hybrid method is considered to consist of two arbitrary but different and linear consistent schemes. Thus the analysis applies for problems where both the PDE and the potentially nonlinear shock-capturing scheme are linearized.
The model problem will be defined in Section 2, and it will be shown that the energy method together with the most natural choice of norm can not be used to prove stability. This does not imply that the energy method categorically fails for this problem, since there are infinitely many energy norms that one could consider.
Thus we instead turn to the Kreiss theory [5], [6] (sometimes referred to as ‘GKS’ theory) and give criteria for which the Kreiss condition is satisfied, thus proving stability. Strikwerda [7] showed that the Kreiss condition leads to stability in the generalized sense for semi-discrete problems. In [6] it was shown that strong stability follows from the Kreiss condition under certain restrictions. The present analysis is somewhat similar to Goldberg and Tadmor [8], who used the Kreiss theory to prove stability of hyperbolic initial-boundary value problems for rather general schemes and boundary conditions. Ciment [9] analyzed a similar coupled problem where both schemes are dissipative.
A few examples will be discussed, including one involving two schemes that are individually stable yet unstable when coupled. This illustrates that stability of hybrid methods is both an important and non-trivial matter.
Section snippets
Problem definition
Consider the hyperbolic half-space problem
Note that while a scalar problem is analyzed here, the analysis applies equally to hyperbolic systems of formwhere A is a diagonalizable matrix. For such systems, the scalar model problem (1) represents each component of the diagonalized system (cf. [8]).
To facilitate later discretization using different schemes coupled at , the model problem can equivalently be written on the folded form
The Kreiss theory
The Kreiss theory [5], [6] finds an energy estimate by use of Laplace transforms in time, and hence it considers normal modes of form and . Inserting into (3a), (3b) yields two constant-coefficient difference equations with solutions of form and where and are roots of the characteristic equations (cf. [6])where . Stability is ensured if the Kreiss condition is satisfied, which amounts to verifying that
Summary and discussion
The case of hybrid finite difference methods, where different schemes are applied in different regions of the domain, is investigated from a stability perspective. The Kreiss theory [5], [6] is used to analyze stability for general linear schemes of arbitrary order and stencil size. The analysis consists of two separate parts, where a limited number of potential multiple roots must be examined individually, whereas the remainder of the half-plane can be handled by finding points where
Acknowledgment
Financial support has been provided by the DoE Scientific Discovery through Advanced Computing (SciDAC) program. The first author also gratefully acknowledges additional support from the Natural Sciences and Engineering Research Council of Canada.
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