Symmetrical weighted essentially non‐oscillatory‐flux limiter schemes for Hamilton–Jacobi equations

Abstract

In this paper, we propose a new scheme that combines weighted essentially non-oscillatory (WENO) procedures together with monotone upwind schemes to approximate the viscosity solution of the Hamilton-Jacobi equations. In onedimensional (1D) case, first, we obtain an optimum polynomial on a four-point stencil. This optimum polynomial is third-order accurate in regions of smoothness. Next, we modify a second-order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten-Osher reconstructionevolution method limiter. Finally, the optimum polynomial is considered as a symmetric and convex combination of three polynomials with ideal weights. Following the methodology of the classic WENO procedure, then, we calculate the nonoscillatory weights with the ideal weights. Numerical experiments in 1D and 2D are performed to compare the capability of the hybrid scheme to WENO schemes.

Arc Length-Based WENO Scheme for Hamilton–Jacobi Equations

“…The physical significance of (1) is important as it appears in several applications such as seismic (1) t ( , t) + H( , t, , D ) = 0, ( , 0) = 0 ( ), waves, image processing, optimal control, calculus of variations, robotic navigation, crystal growth, etching, differential games and geometric optics. The difficulty in dealing with (1) is that it develops the discontinuous derivatives even with smooth initial data. As a result, the solutions for (1) are not available as a unique sense and to further study solutions are understood in a weaker sense.…”

“…The difficulty in dealing with (1) is that it develops the discontinuous derivatives even with smooth initial data. As a result, the solutions for (1) are not available as a unique sense and to further study solutions are understood in a weaker sense. Weak solutions were introduced by the notion of viscosity solutions [10,12,13].…”

“…It is a well-known fact (see [11,13]) that (1) has a close connection to the conservation laws in its structure. As an immediate consequence, the successful numerical methods for conservation laws can be adapted intently for solving (1). In such direction, Osher and Sethian [22] constructed a second-order essentially non-oscillatory (ENO) scheme, and Osher and Shu [23] presented higher order ENO schemes.…”

“…Further, the WENO schemes come into the literature to solve these equations, and many higher order WENO variants developed, such as Hermite WENO schemes on structured meshes by Qiu [24,25], Qiu and Shu [26] and Zheng and Qiu [32]; on unstructured meshes Lafon and Osher [19], Abgrall [2], Augoula and Abgrall [3], Zhang and Shu [31], Li and Chan [20]. Few more notable WENO schemes are central WENO schemes [6], weighted power ENO schemes [28], mapped WENO schemes [7,15], WENO-L 1 [27], symmetrical WENO schemes [1], and WENO-ZQ scheme [34]. There are also central high-resolution schemes by Kurgunov and Tadmor [18], Lin and Tadmor [21], Bryson and Levy [5].…”

“…Such developments can be seen in Ref. [4] for the finite volume formulation for conservation laws; here, we introduce such a concept in the finite difference approach and extend to solve (1). These smoothness indicators approximate the derivatives involved in the reconstructed polynomials effectively with the curve length-based measurements.…”