A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations
A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations
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DOI：
10.1016/j.cma.2020.113155
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发表时间：
2020-08
影响因子：
7.2
通讯作者：
T. Arbogast;Chieh-Sen Huang;X. Zhao;Danielle N. King
中科院分区：
文献类型：
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作者：
T. Arbogast;Chieh-Sen Huang;X. Zhao;Danielle N. King
给出了标量双曲守恒律或对流扩散方程的有限体积近似。在直线法的背景下，空间离散使用加权基本无振荡（韦诺）重建与自适应阶（WENO-AO），和时间演化使用隐式龙格库塔方法。因此，时间步长可以大于CFL时间步长。为了减少振荡的解决方案，空间分区龙格库塔方法的相关思想被使用。自适应龙格-库塔法的发展，融合了L-稳定，三阶，隐式Radau IIA方法与复合向后欧拉方法使用的加权程序的启发，从空间韦诺方法。加权过程需要平滑度指示器，并且考虑了几种可能性，尽管其中一种可能被认为是优选的。整个计划被证明保持三阶精度时，解决方案是光滑的。当解有间断时，计算表明该方案在远离激波处具有三阶精度，并达到向后欧拉方法的总体精度。数值算例表明，自适应龙格-库塔法减少了解的振荡。此外，所得到的计划是无条件L-稳定的光滑解的线性问题。
A finite volume approximation of the scalar hyperbolic conservation law or advection–diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge–Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge–Kutta methods are used. An adaptive Runge–Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge–Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem.