一种可求解混合型双曲守恒律方程的高阶数值格式

A High Order Numerical Scheme for Solving Conservation Laws of Mixed Type

摘要: 在计算流体力学等领域中，双曲守恒方程及其对流占优问题可以通过加权基本无振荡(WENO)方法进行高精度的数值求解。本文旨在通过建立一个修正高阶有限差分WENO格式来求解双曲–椭圆混合型方程。通过引入特殊的通量分裂方法将通量分解为两部分，在这两个分量中分别应用双曲WENO算子，并在守恒律方程数值通量中加入高阶修正项，获得了一种可求解混合型守恒律方程的高精度有限差分WENO格式。该离散格式主要用于求解viscosity-capillarity容许性条件下的双曲–椭圆型范德华方程。数值测试验证了该算法的高精度和有效性，结果表明，该格式不仅能在强间断区域保持无振荡，在解的光滑部分保持高阶数值精度，还可以有效描述复杂波结构。

Abstract: Hyperbolic conservation law equations and convection-dominant situations in computational fluid dynamics and other areas can be solved numerically with great precision using the weighted essentially non-oscillatory (WENO) techniques. In this paper, we attempt to address the hyperbolic-elliptic mixed equations by developing a corrected high-order finite difference WENO scheme. By introducing a special flux splitting method, the flux is decomposed into two parts. We then apply the hyperbolic WENO operator to these two components, and add the higher order correction term to the numerical flux of the conservation laws, and finally a high-precision finite-difference WENO scheme is obtained. The discretization scheme is mainly used to solve the hyperbolic-elliptic van der Waals equations under the viscosity-capillarity admissibility criterion. It can be shown by numerical examples that the scheme not only can preserve the necessary no oscillation in the discontinuous region, but also retain high order numerical accuracy in the smooth part of the solution, and can effectively describe the complex wave structure.