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Adaptive Multiresolution Methods for the Simulation of Waves in Excitable Media

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Abstract

We present fully adaptive multiresolution methods for a class of spatially two-dimensional reaction-diffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spatial coherence and serves as a mechanism for obtaining sharper wave fronts. The multiresolution method is formulated on the basis of two alternative reference schemes, namely a classical finite volume method, and Barkley’s approach (Barkley in Phys. D 49:61–70, 1991), which consists in separating the computation of the nonlinear reaction terms from that of the piecewise linear diffusion. The proposed methods are enhanced with local time stepping to attain local adaptivity both in space and time. The computational efficiency and the numerical precision of our methods are assessed. Results illustrate that the fully adaptive methods provide stable approximations and substantial savings in memory storage and CPU time while preserving the accuracy of the discretizations on the corresponding finest uniform grid.

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Authors and Affiliations

  1. CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Raimund Bürger

  2. Modelling and Scientific Computing, CMCS-MATHICSE-SB, École Polytechnique Fédérale de Lausanne, EPFL, Station 8, 1015, Lausanne, Switzerland

    Ricardo Ruiz-Baier

  3. Centre de Mathématiques et d’Informatique, Université de Provence, 39 rue Joliot-Curie, 13453, Marseille cedex 13, France

    Kai Schneider

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  1. Raimund Bürger
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  3. Kai Schneider
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Correspondence to Ricardo Ruiz-Baier.

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Bürger, R., Ruiz-Baier, R. & Schneider, K. Adaptive Multiresolution Methods for the Simulation of Waves in Excitable Media. J Sci Comput 43, 261–290 (2010). https://doi.org/10.1007/s10915-010-9356-3

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