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The Flux Reconstruction Method with Lax–Wendroff Type Temporal Discretization for Hyperbolic Conservation Laws

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Abstract

In this paper, we develop a Lax–Wendroff type time discretization method for high order Flux Reconstruction scheme to solve hyperbolic conservation laws. Through Cauchy–Kowalewski procedure, the resulting Lax–Wendroff Flux Reconstruction (LWFR) scheme is an alternative spatial–temporal coupling method to the popular Runge–Kutta Flux Reconstruction (RKFR) scheme. LWFR is a one-step explicit high order discontinuous finite element method and its discretization procedure is more compact and effective for certain problems than that of RKFR. Furthermore, aiming at accurate simulation of discontinuity, we propose a robust local artificial viscosity formulation of LWFR for the first time. A collection of successful numerical experiments show that LWFR can give essentially non-oscillatory and sharp solutions for discontinuity, and maintain designed order of accuracy for smooth regions, both in one-dimensional and two-dimensional Euler equations. In conclusion, LWFR scheme is cost effective and accuracy-preserving for certain problems and can be an attractive candidate to solve hyperbolic conservation laws.

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Acknowledgements

This research was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11721202 and 11972064).

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Appendix A

Appendix A

The details of the formulation in Sect. 2.4 Eq. (32) are as following:

$$ \begin{aligned} {\mathbf{S}}_{{\mathbf{A}}} & = \left( {{\mathbf{S}}_{{{\mathbf{A}}1}}^{T} ,{\mathbf{S}}_{{{\mathbf{A}}2}}^{T} ,{\mathbf{S}}_{{{\mathbf{A}}3}}^{T} ,{\mathbf{S}}_{{{\mathbf{A}}4}}^{T} } \right)^{T} \\ {\mathbf{S}}_{{{\mathbf{A}}k}} & = \left[ {sa_{m,n}^{k} } \right]_{4 \times 4} ,sa_{m,n}^{k} = \frac{{\partial a_{k,m} }}{{\partial q_{n} }},\quad k ,m,n = 1,2,3,4 \\ \end{aligned} $$
(51)
$$ \begin{aligned} {\mathbf{S}}_{{\mathbf{B}}} & = \left( {{\mathbf{S}}_{{{\mathbf{B}}1}}^{T} ,{\mathbf{S}}_{{{\mathbf{B}}2}}^{T} ,{\mathbf{S}}_{{{\mathbf{B}}3}}^{T} ,{\mathbf{S}}_{{{\mathbf{B}}4}}^{T} } \right)^{T} \\ {\mathbf{S}}_{{{\mathbf{B}}k}} & = \left[ {sb_{m,n}^{k} } \right]_{4 \times 4} ,sb_{m,n}^{k} = \frac{{\partial b_{k,m} }}{{\partial q_{n} }},\quad k ,m,n = 1,2,3,4 \\ \end{aligned} $$
(52)

where \( a_{k,m} \) and \( b_{k,m} \) are the \( m \)th component of vector \( {\mathbf{a}}_{k} \) and \( {\mathbf{b}}_{k} \), respectively.

$$ {\mathbf{SA}}_{1} = {\mathbf{0}} $$
(53)
$$ {\mathbf{SA}}_{2} = \left[ {\begin{array}{*{20}c} {\left( {3 - \gamma } \right)\frac{{q_{2}^{2} }}{{q_{1}^{3} }} - \left( {\gamma - 1} \right)\frac{{q_{3}^{2} }}{{q_{1}^{3} }}} & { - \left( {3 - \gamma } \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & {\left( {\gamma - 1} \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & 0 \\ { - \left( {3 - \gamma } \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & {\frac{{\left( {3 - \gamma } \right)}}{{q_{1} }}} & 0 & 0 \\ {\left( {\gamma - 1} \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & 0 & { - \frac{{\left( {\gamma - 1} \right)}}{{q_{1} }}} & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
(54)
$$ {\mathbf{SA}}_{3} = \left[ {\begin{array}{*{20}c} {\frac{{2q_{2} q_{3} }}{{q_{1}^{3} }}} & { - \frac{{q_{3} }}{{q_{1}^{2} }}} & { - \frac{{q_{2} }}{{q_{1}^{2} }}} & 0 \\ { - \frac{{q_{3} }}{{q_{1}^{2} }}} & 0 & {\frac{1}{{q_{1} }}} & 0 \\ { - \frac{{q_{2} }}{{q_{1}^{2} }}} & {\frac{1}{{q_{1} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
(55)
$$ {\mathbf{SA}}_{4} = \left[ {\begin{array}{*{20}c} { - 3\left( {\gamma - 1} \right)\frac{{q_{2}^{3} + q_{2} q_{3}^{2} }}{{q_{1}^{4} }} + 2\gamma \frac{{q_{2} q_{4} }}{{q_{1}^{3} }}} & {\left( {\gamma - 1} \right)\frac{{3q_{2}^{2} + q_{3}^{2} }}{{q_{1}^{3} }} - \gamma \frac{{q_{4} }}{{q_{1}^{2} }}} & {2\left( {\gamma - 1} \right)\frac{{q_{2} q_{3} }}{{q_{1}^{3} }}} & { - \gamma \frac{{q_{2} }}{{q_{1}^{2} }}} \\ {\left( {\gamma - 1} \right)\frac{{3q_{2}^{2} + q_{3}^{2} }}{{q_{1}^{3} }} - \gamma \frac{{q_{4} }}{{q_{1}^{2} }}} & { - 3\left( {\gamma - 1} \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & { - \left( {\gamma - 1} \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & {\frac{\gamma }{{q_{1} }}} \\ {2\left( {\gamma - 1} \right)\frac{{q_{2} q_{3} }}{{q_{1}^{3} }}} & { - \left( {\gamma - 1} \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & { - \left( {\gamma - 1} \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & 0 \\ { - \gamma \frac{{q_{2} }}{{q_{1}^{2} }}} & {\frac{\gamma }{{q_{1} }}} & 0 & 0 \\ \end{array} } \right] $$
(56)
$$ {\mathbf{SB}}_{1} = {\mathbf{0}} $$
(57)
$$ {\mathbf{SB}}_{2} = \left[ {\begin{array}{*{20}c} {\frac{{2q_{2} q_{3} }}{{q_{1}^{3} }}} & { - \frac{{q_{3} }}{{q_{1}^{2} }}} & { - \frac{{q_{2} }}{{q_{1}^{2} }}} & 0 \\ { - \frac{{q_{3} }}{{q_{1}^{2} }}} & 0 & {\frac{1}{{q_{1} }}} & 0 \\ { - \frac{{q_{2} }}{{q_{1}^{2} }}} & {\frac{1}{{q_{1} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
(58)
$$ {\mathbf{SB}}_{3} = \left[ {\begin{array}{*{20}c} {\left( {3 - \gamma } \right)\frac{{q_{3}^{2} }}{{q_{1}^{3} }} - \left( {\gamma - 1} \right)\frac{{q_{2}^{2} }}{{q_{1}^{3} }}} & {\left( {\gamma - 1} \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & { - \left( {3 - \gamma } \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & 0 \\ {\left( {\gamma - 1} \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & { - \frac{{\left( {\gamma - 1} \right)}}{{q_{1} }}} & 0 & 0 \\ { - \left( {3 - \gamma } \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & 0 & {\frac{{\left( {3 - \gamma } \right)}}{{q_{1} }}} & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
(59)
$$ {\mathbf{SB}}_{4} = \left[ {\begin{array}{*{20}c} { - 3\left( {\gamma - 1} \right)\frac{{q_{2}^{2} q_{3} + q_{3}^{3} }}{{q_{1}^{4} }} + 2\gamma \frac{{q_{3} q_{4} }}{{q_{1}^{3} }}} & {2\left( {\gamma - 1} \right)\frac{{q_{2} q_{3} }}{{q_{1}^{3} }}} & {\left( {\gamma - 1} \right)\frac{{q_{2}^{2} + 3q_{3}^{2} }}{{q_{1}^{3} }} - \gamma \frac{{q_{4} }}{{q_{1}^{2} }}} & { - \gamma \frac{{q_{3} }}{{q_{1}^{2} }}} \\ {2\left( {\gamma - 1} \right)\frac{{q_{2} q_{3} }}{{q_{1}^{3} }}} & { - \left( {\gamma - 1} \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & { - \left( {\gamma - 1} \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & 0 \\ {\left( {\gamma - 1} \right)\frac{{q_{2}^{2} + 3q_{3}^{2} }}{{q_{1}^{3} }} - \gamma \frac{{q_{4} }}{{q_{1}^{2} }}} & { - \left( {\gamma - 1} \right)\frac{{q_{2} }}{{q_{1}^{2} }}} & { - 3\left( {\gamma - 1} \right)\frac{{q_{3} }}{{q_{1}^{2} }}} & {\frac{\gamma }{{q_{1} }}} \\ { - \gamma \frac{{q_{3} }}{{q_{1}^{2} }}} & 0 & {\frac{\gamma }{{q_{1} }}} & 0 \\ \end{array} } \right] $$
(60)

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Lou, S., Yan, C., Ma, LB. et al. The Flux Reconstruction Method with Lax–Wendroff Type Temporal Discretization for Hyperbolic Conservation Laws. J Sci Comput 82, 42 (2020). https://doi.org/10.1007/s10915-020-01146-8

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  • DOI: https://doi.org/10.1007/s10915-020-01146-8

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