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Construction of Convergent High Order Schemes for Time Dependent Hamilton–Jacobi Equations

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Abstract

We construct a class of convergent high order schemes for time dependent Hamilton–Jacobi equations. In general, high order schemes such as WENO scheme achieve high order accuracy in smooth regions of the solution and an essentially non-oscillatory resolution at singularities of the solution, but despite its good numerical properties, the convergence to the viscosity solution could not be expected for certain nonconvex problems. We propose a general method of constructing convergent high order schemes and discuss the question of its convergence. The scheme relies on the reasonable combination of a high order scheme and a first order monotone scheme, which is determined so as to make the scheme converge while achieving high order accuracy. We provide adaptive algorithms for problems with nonconvex Hamiltonians and perform a detailed numerical study to demonstrate its convergence.

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Kwangil Kim & Yonghai Li

  2. Department of Mathematics, University of Science, Pyongyang, DPR, Korea

    Kwangil Kim

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  1. Kwangil Kim
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  2. Yonghai Li
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Correspondence to Kwangil Kim.

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Kim, K., Li, Y. Construction of Convergent High Order Schemes for Time Dependent Hamilton–Jacobi Equations. J Sci Comput 65, 110–137 (2015). https://doi.org/10.1007/s10915-014-9955-5

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  • DOI: https://doi.org/10.1007/s10915-014-9955-5

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