一维双曲守恒律方程的保极值无振荡杂交有限体积格式

An Extrema-Preserving Hybrid Non-Oscillatory Finite Volume Scheme for One-Dimensional Hyperbolic Conservation Laws

摘要: 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。

Abstract: 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.