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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 W. E. and 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:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

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