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High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws

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Abstract

Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large class of time integrators include all popular (multistage) Runge–Kutta as well as single-step (multiderivative) Taylor methods. (The latter are commonly referred to as Lax–Wendroff methods when applied to PDEs). In this work, we offer explicit multistage multiderivative time integrators for hyperbolic conservation laws. Like Lax–Wendroff methods, multiderivative integrators permit the evaluation of higher derivatives of the unknown in order to decrease the memory footprint and communication overhead. Like traditional Runge–Kutta methods, multiderivative integrators admit the addition of extra stages, which introduce extra degrees of freedom that can be used to increase the order of accuracy or modify the region of absolute stability. We describe a general framework for how these methods can be applied to two separate spatial discretizations: the discontinuous Galerkin (DG) method and the finite difference essentially non-oscillatory (FD-WENO) method. The two proposed implementations are substantially different: for DG we leverage techniques that are closely related to generalized Riemann solvers; for FD-WENO we construct higher spatial derivatives with central differences. Among multiderivative time integrators, we argue that multistage two-derivative methods have the greatest potential for multidimensional applications, because they only require the flux function and its Jacobian, which is readily available. Numerical results indicate that multiderivative methods are indeed competitive with popular strong stability preserving time integrators.

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Notes

  1. When applied to partial differential equations, Taylor methods are commonly referred to as Lax–Wendroff methods.

  2. A finite difference method is conservative if the method satisfies \(\frac{d}{dt}\left( \sum _i q_i(t) \right) = 0\) on a periodic (or infinite) domain.

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Acknowledgments

This work has been supported in part by Air Force Office of Scientific Research grants FA9550-11-1-0281, FA9550-12-1-0343 and FA9550-12-1-0455, and by National Science Foundation Grant number DMS-1115709. We would like to thank Matthew F. Causley for discussing multiderivative methods with us, and Qi Tang for useful discussions on the WENO method.

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  1. Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI, 48824, USA

    David C. Seal & Yaman Güçlü

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Andrew J. Christlieb

  3. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Andrew J. Christlieb

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  1. David C. Seal
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Seal, D.C., Güçlü, Y. & Christlieb, A.J. High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws. J Sci Comput 60, 101–140 (2014). https://doi.org/10.1007/s10915-013-9787-8

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