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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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