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Enslaved finite difference schemes for nonlinear dissipative PDEs. (English) Zbl 0879.65063

The authors describe a method to increase the accuracy and numerical efficiency of a given scheme for the solution of nonlinear dissipative partial differential equations (PDEs). The solution is studied on two finite difference meshes, a coarse one and a fine one. Under the assumption that time derivatives are smaller than the dominant terms in the PDE, coupled equations for the evolution of both low-order and high-order modes of the solution are formulated in terms of the coarse grid. An algebraic relation between resolved and unresolved scales of motion is viewed as an enslavement of the smaller, higher-order modes. The enslaved scheme reaches the accuracy of a standard one on a mesh twice as fine without dictating a time step decrease corresponding to such a mesh.
The general concept is studied in detail for the Burgers equation as a prototype of advection/diffusion equations. Several numerical tests are carried out. The modified scheme includes a function \(\Phi\) which estimates the effects of the unresolved scales on the coarse grid. The most difficult part of the method is to calculate this enslaving function \(\Phi\), which is done by use of a contraction mapping. Whether the modified scheme is more efficient than the original one depends on the trade-off between improved accuracy and increased expense. For small time derivatives of the solution, significant improvements are expected and confirmed by several computational examples.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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