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Finite difference heterogeneous multi-scale method for homogenization problems. (English) Zbl 1034.65067
Authors’ abstract: We propose a numerical method, the finite difference heterogeneous multi-scale method, for solving multi-scale parabolic problems. Based on the framework introduced by {\it W. E.} and {\it B. Engquist} [The heterogeneous multi-scale methods, Commun. Math. Sci. 1, 87--132 (2003)], the numerical method relies on the use of two different schemes for the original equation, at different grid level which allows to give numerical results at a much lower cost than solving the original equations. We describe the strategy for constructing such a method, discuss generalization for cases with time dependency, random correlated coefficients, non-conservative form and implementation issues. Finally, the new method is illustrated with several test examples.

MSC:
65M06Finite difference methods (IVP of PDE)
35K15Second order parabolic equations, initial value problems
35B27Homogenization; equations in media with periodic structure (PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
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References:
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