Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations
Kernel Based High Order “Explicit” Unconditionally Stable Scheme for Nonlinear Degenerate Advection-Diffusion Equations
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DOI：
10.1007/s10915-020-01152-w
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发表时间：
2017-07
影响因子：
2.5
通讯作者：
A. Christlieb;Wei Guo;Yan Jiang;Hyoseon Yang
中科院分区：
文献类型：
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作者：
A. Christlieb;Wei Guo;Yan Jiang;Hyoseon Yang
本文给出了一类具有非光滑解的非线性退化抛物型方程的一种新的数值格式。所提出的方法依赖于在我们早期关于直线转置和逐次卷积方法的工作中找到的解的特殊的基于核的公式。在该框架中，进一步使用了高阶加权基本无振荡方法和非线性滤波器来避免虚假振荡。利用高阶显式强保稳龙格-库塔方法实现了时间上的高阶精度。此外，基于核的格式与显式SSP方法相结合的理论研究表明，组合格式是无条件稳定的，并且可以达到三阶精度。使用快速求和算法对基于核的方法进行了评估。与其他具有相同阶次精度的显式格式相比，新方法允许更大的时间步长演化，从而显著地节省了计算。
In this paper, we present a novel numerical scheme for solving a class of nonlinear degenerate parabolic equations with non-smooth solutions. The proposed method relies on a special kernel based formulation of the solutions found in our early work on the method of lines transpose and successive convolution. In such a framework, a high order weighted essentially non-oscillatory methodology and a nonlinear filter are further employed to avoid spurious oscillations. High order accuracy in time is realized by using the high order explicit strong-stability-preserving (SSP) Runge-Kutta method. Moreover, theoretical investigations of the kernel based formulation combined with an explicit SSP method indicate that the combined scheme is unconditionally stable and up to third order accuracy. Evaluation of the kernel based approach is done with a fastsummation algorithm. The new method allows for much larger time step evolution compared with other explicit schemes with the same order accuracy, leading to remarkable computational savings.