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Unconditional stability of corrected explicit-implicit domain decomposition algorithms for parallel approximation of heat equations. (English) Zbl 1125.65087

Explicit-implicit domain decomposition (EIDD) methods are computationally and communicationally efficient for each time step but always suffer from small step size restrictions. In this paper, the authors present a class of corrected explicit-implicit domain decomposition (CEIDD) methods for the parallel approximation of the linear heat equations. By adding an interface correction step to Kuznetsov’s EIDD, the one-dimensional CEIDD procedure achieves unconditionally stability without discarding the time-stepwise efficiency of the EIDD method.
For multidimensional problems, special zigzag-shaped interfaces are suggested in the CEIDD method in order to maintain the virtues of CEIDD method and improve the flexibility in domain partitioning. Based on non-crossover and crossover types of the zigzag interface, the resulting CEIDD-ZI algorithms are proposed for two strategies of a subdomain partition. By the energy method, it is shown that the proposed algorithms, including their degenerate cases – the corrected explicit hopscotch schemes, are convergent in the discrete \(H^{1}\) semi-norm and \(L^{2}\) norm. Numerical experiments confirm the results in the theoretical analysis.

MSC:

65M12 Stability and convergence of numerical 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
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65Y05 Parallel numerical computation
35K05 Heat equation
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