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A sub-cell WENO reconstruction method for spatial derivatives in the ADER scheme. (English) Zbl 1349.65279
Summary: We introduce a sub-cell WENO reconstruction method to evaluate spatial derivatives in the high-order ADER scheme. The basic idea in our reconstruction is to use only \(r\) stencils to reconstruct the point-wise values of solutions and spatial derivatives for the \((2r-1)\)th-order ADER scheme in one dimension, while in two dimensions, the dimension-by-dimension sub-cell reconstruction approach for spatial derivatives is employed.compared with the original ADER scheme of Toro and Titarev (2002) [2] that uses the direct derivatives of reconstructed polynomials for solutions to evaluate spatial derivatives, our method not only reduces greatly the computational costs of the ADER scheme on a given mesh, but also avoids possible numerical oscillations near discontinuities, as demonstrated by a number of one- and two-dimensional numerical tests. All these tests show that the 5th-order ADER scheme based on our sub-cell reconstruction method achieves the desired accuracy, and is essentially non-oscillatory and computationally cheaper for problems with discontinuities.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
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[1] Toro, E. F.; Millington, R. C.; Nejad, L. A.M., Towards very high order Godunov schemes, (Toro, E. F., Godunov Methods. Theory and Applications, Edited Review, (2001), Kluwer/Plenum Academic Publishers Dordrecht), 907-940 · Zbl 0989.65094
[2] Toro, E. F.; Titarev, V. A., Solution of the generalised Riemann problem for advection-reaction equations, Proc. R. Soc. London, Ser. A, 458, 271-281, (2002) · Zbl 1019.35061
[3] Schwartzkopf, T.; Munz, C. D.; Toro, E. F., ADER: a high-order approach for linear hyperbolic systems in 2D, J. Sci. Comput., 17, 231-240, (2002) · Zbl 1022.76034
[4] Titarev, V. A.; Toro, E. F., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 609-618, (2002) · Zbl 1024.76028
[5] Titarev, V. A.; Toro, E. F., High-order ADER schemes for scalar advection reaction-diffusion equations, CFD J., 12, 1-6, (2003)
[6] Toro, E. F.; Titarev, V. A., TVD fluxes for the high-order ADER schemes, J. Sci. Comput., 24, 285-309, (2005) · Zbl 1096.76029
[7] Toro, E. F.; Titarev, V. A., ADER schemes for scalar hyperbolic conservation laws in three space dimensions, J. Comput. Phys., 202, 196-215, (2005) · Zbl 1061.65103
[8] Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional non-linear hyperbolic systems, J. Comput. Phys., 204, 715-736, (2005) · Zbl 1060.65641
[9] Dumbser, M.; Balsara, D.; Toro, E. F.; Munz, C. D., A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 8209-8253, (2008) · Zbl 1147.65075
[10] Hidalgo, A.; Dumbser, M., ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations, J. Sci. Comput., 48, 173-189, (2011) · Zbl 1221.65231
[11] Montecinos, G. I.; Castro, C. E.; Dumbser, M.; Toro, E. F., Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms, J. Comput. Phys., (2012) · Zbl 1284.35268
[12] Dumbser, M.; Enaux, C.; Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227, 3971-4001, (2008) · Zbl 1142.65070
[13] Jiang, G. S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228, (1996) · Zbl 0877.65065
[14] Shi, J.; Hu, C.; Shu, C.-W., A technique of treating negative weights in WENO schemes, J. Comput. Phys., 175, 108-127, (2002) · Zbl 0992.65094
[15] Balsara, D. S.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452, (2000) · Zbl 0961.65078
[16] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes III, J. Comput. Phys., 71, 231-303, (1987) · Zbl 0652.65067
[17] Casper, J.; Atkins, H., A finite-volume high order ENO scheme for two dimensional hyperbolic systems, J. Comput. Phys., 106, 62-76, (1993) · Zbl 0774.65066
[18] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[19] Woodwardand, P.; Colella, P., The numerical simulation of two dimensional fluids with strong shock, J. Comput. Phys., 54, 115-173, (1984)
[20] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Verlag · Zbl 0923.76004
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