A New Type of Finite Volume WENO Schemes for Hyperbolic Conservation Laws

AbstractA new type of finite difference weighted essentially non-oscillatory (WENO) schemes for hyperbolic conservation laws was designed in Zhu and Qiu (J Comput Phys 318:110–121, 2016), in this continuing paper, we extend such methods to finite volume version in multi-dimensions. There are two major advantages of the new WENO schemes superior to the classical finite volume WENO schemes (Shu, in: Quarteroni (ed) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, CIME subseries, Springer, Berlin, 1998), the first is the associated linear weights can be any positive numbers with only requirement that their summation equals one, and the second is their simplicity and easy extension to multi-dimensions in engineering applications. The new WENO reconstruction is a convex combination of a fourth degree polynomial with two linear polynomials defined on unequal size spatial stencils in a traditional WENO fashion. These new fifth order WENO schemes use the same number of cell average information as the classical fifth order WENO schemes Shu (1998), could get less absolute numerical errors than the classical same order WENO schemes, and compress nonphysical oscillations nearby strong shocks or contact discontinuities. Some benchmark tests are performed to illustrate the capability of these schemes.

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