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Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations

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Abstract

In this paper, a class of high-order central Hermite WENO (HWENO) schemes based on finite volume framework and staggered meshes is proposed for directly solving one- and two-dimensional Hamilton-Jacobi (HJ) equations. The methods involve the Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. This work can be regarded as an extension of central HWENO schemes for hyperbolic conservation laws (Tao et al. J. Comput. Phys. 318, 222–251, 2016) which combine the central scheme and the HWENO spatial reconstructions and therefore carry many features of both schemes. Generally, it is not straightforward to design a finite volume scheme to directly solve HJ equations and a key ingredient for directly solving such equations is the reconstruction of numerical Hamiltonians to guarantee the stability of methods. Benefited from the central strategy, our methods require no numerical Hamiltonians. Meanwhile, the zeroth-order and the first-order moments of the solution are involved in the spatial HWENO reconstructions which is more compact compared with WENO schemes. The reconstructions are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. A collection of one- and two- dimensional numerical examples is performed to validate high resolution and robustness of the methods in approximating the solutions of HJ equations, which involve linear, nonlinear, smooth, non-smooth, convex or non-convex Hamiltonians.

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Author notes

  1. Zhanjing Tao

    Present address: Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian, 361005, People’s Republic of China

    Zhanjing Tao & Jianxian Qiu

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  1. Zhanjing Tao
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  2. Jianxian Qiu
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Correspondence to Jianxian Qiu.

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Communicated by: Helge Holden

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Tao, Z., Qiu, J. Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton-Jacobi equations. Adv Comput Math 43, 1023–1058 (2017). https://doi.org/10.1007/s10444-017-9515-2

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  • DOI: https://doi.org/10.1007/s10444-017-9515-2

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