Abstract 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.

中文翻译：

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对流-扩散方程的三阶、隐式、有限体积、自适应 Runge-Kutta WENO 格式

摘要 给出了标量双曲守恒定律或对流扩散方程的有限体积近似。在直线法的上下文中，空间离散化使用加权的本质非振荡 (WENO) 重构和自适应阶数 (WENO-AO)，时间演化使用隐式 Runge-Kutta 方法。因此时间步长可能大于 CFL 时间步长。为了减少解中的振荡，使用了与空间分区 Runge-Kutta 方法相关的想法。开发了一种自适应 Runge-Kutta 方法，该方法使用受空间 WENO 方法启发的加权程序将 L 稳定、三阶、隐式 Radau IIA 方法与复合后向欧拉方法混合。加权过程需要一个平滑度指标，并考虑了几种可能性，虽然一个人可能被认为是首选。整体方案被证明在解决方案平滑时保持三阶精度。当解具有不连续性时，该方案在计算上显示为远离冲击的三阶精度，并达到后向欧拉方法的整体精度。数值例子表明，自适应 Runge-Kutta 方法减少了解中的振荡。此外，对于线性问题的平滑解，结果方案被证明是无条件 L 稳定的。数值例子表明自适应 Runge-Kutta 方法减少了解中的振荡。此外，对于线性问题的平滑解，结果方案被证明是无条件 L 稳定的。数值例子表明，自适应 Runge-Kutta 方法减少了解中的振荡。此外，对于线性问题的平滑解，结果方案被证明是无条件 L 稳定的。