物理系统中波动、传播等现象通常用双曲型守恒律方程的数学模型来描述,特别是在流体力学领域尤为重要。针对此类方程,我们考虑了Lax-Wendroff型中心间断伽辽金方法。该方法首先采用Lax-Wendroff型时间离散方法,也就是通过泰勒级数展开...物理系统中波动、传播等现象通常用双曲型守恒律方程的数学模型来描述,特别是在流体力学领域尤为重要。针对此类方程,我们考虑了Lax-Wendroff型中心间断伽辽金方法。该方法首先采用Lax-Wendroff型时间离散方法,也就是通过泰勒级数展开处理时间导数,然后在空间上运用中心间断伽辽金方法,从而避免了传统的多步时间积分方法。最后我们对多个双曲型守恒律方程开展数值实验,验证所提出方法在计算效率和精度上的有效性。In physical systems, phenomena like wave fluctuation and propagation are often described using hyperbolic conservation law equations, which play a crucial role in fluid mechanics. To solve these equations, we employ the Lax-Wendroff central discontinuous Galerkin method. This approach begins with the Lax-Wendroff time discretization, where time derivatives are managed through a Taylor series expansion. It then incorporates the central discontinuous Galerkin method for spatial discretization and effectively eliminates the need for traditional multi-step time integration schemes. Finally, numerical experiments on various hyperbolic conservation law equations are constructed to validate the effectiveness of our method in terms of both computational efficiency and accuracy.展开更多
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...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.展开更多
文摘物理系统中波动、传播等现象通常用双曲型守恒律方程的数学模型来描述,特别是在流体力学领域尤为重要。针对此类方程,我们考虑了Lax-Wendroff型中心间断伽辽金方法。该方法首先采用Lax-Wendroff型时间离散方法,也就是通过泰勒级数展开处理时间导数,然后在空间上运用中心间断伽辽金方法,从而避免了传统的多步时间积分方法。最后我们对多个双曲型守恒律方程开展数值实验,验证所提出方法在计算效率和精度上的有效性。In physical systems, phenomena like wave fluctuation and propagation are often described using hyperbolic conservation law equations, which play a crucial role in fluid mechanics. To solve these equations, we employ the Lax-Wendroff central discontinuous Galerkin method. This approach begins with the Lax-Wendroff time discretization, where time derivatives are managed through a Taylor series expansion. It then incorporates the central discontinuous Galerkin method for spatial discretization and effectively eliminates the need for traditional multi-step time integration schemes. Finally, numerical experiments on various hyperbolic conservation law equations are constructed to validate the effectiveness of our method in terms of both computational efficiency and accuracy.
文摘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.
