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。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.
本文提出一种保极值高分辨率杂交有限体积格式数值求解一维双曲守恒律方程。基于对流有界准则和TVD准则,并结合Hermite插值过程构造新的高分辨率格式。为克服TVD性质导致的非单调光滑解精度损失,构造杂交因子来有效地识别光滑和间断区域,从而形成杂交高分辨率格式。关于时间积分的常微分方程组使用3阶Runge-Kutta格式进行数值求解。典型数值算例结果显示杂交格式在解的光滑极值点处能保持与线性高阶格式相同的高精度,有效克服了光滑极值点的精度损失而且在间断附近能够有效的抑制非物理振荡。In this paper, an extrema-preserving hybrid non-oscillatory finite volume scheme is proposed to numerically solve the one-dimensional hyperbolic conservation laws. A new high-resolution scheme is constructed based on the convection boundedness criteria CBC and the TVD criterion and a Hermite interpolation process. In order to overcome the loss of accuracy of non-monotonic smooth solutions caused by the nature of TVD, hybrid indicator is constructed to effectively identify smooth and discontinuous regions, so as to form a hybrid high-resolution scheme. Systems of ordinary differential equations about time integration are solved numerically using the third-order Runge-Kutta format. Numerical experiments on typical test cases show that the hybrid scheme achieves third-order accuracy at the smooth extremum of the solution and effectively suppress unphysical oscillations in the vicinity of discontinuities.
