A high-order symmetrical weighted hybrid ENO-flux limiter scheme for hyperbolic conservation laws

Abstract In this paper, we propose a new weighted essentially non-oscillatory (WENO) procedure for solving hyperbolic conservation laws, on uniform meshes. The new scheme combines essentially non-oscillatory (ENO) reconstructions together with monotone upwind schemes for scalar conservation laws’ interpolants. In a one-dimensional context, first, we obtain an optimum polynomial on a five-cells stencil. This optimum polynomial is fifth-order accurate in regions of smoothness. Next, we modify a third-order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction–evolution method limiter. Then, we consider the optimum polynomial as a symmetric and convex combination of four polynomials with ideal weights. After that, following the methodology of the classic WENO procedure, we calculate non-oscillatory weights with the ideal weights. Also, the numerical solution is advanced in time by means of the linear multi-step total variation bounded ( TV B 0 ) technique. Numerical examples on both scalar and gas dynamics problems confirm that the new scheme is non-oscillatory and yields sharp results when solving profiles with discontinuities. Comparing the new scheme with high-order WENO schemes shows that our method reduces smearing near shocks and corners, and in some cases it is more accurate near discontinuities. Finally, the new method is extended to multi-dimensional problems by a dimension-by-dimension approach. Several multi-dimensional examples are performed to show that our method remains non-oscillatory while giving good resolution of discontinuities.

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