Semi-implicit spectral deferred correction (SDC) methods provide a systematic approach to construct time integration methods of arbitrarily high order for nonlinear evolution equations including conservation laws. They converge towards $A$- or even $L$-stable collocation methods, but are often not sufficiently robust themselves. In this paper, a family of SDC methods inspired by an implicit formulation of the Lax-Wendroff method is developed. Compared to fully implicit approaches, the methods have the advantage that they only require the solution of positive definite or semi-definite linear systems. Numerical evidence suggests that the proposed semi-implicit SDC methods with Radau points are $L$-stable up to order 11 and require very little diffusion for orders 13 and 15. The excellent stability and accuracy of these methods is confirmed by numerical experiments with 1D conservation problems, including the convection-diffusion, Burgers, Euler and Navier-Stokes equations.

翻译：半隐式谱延迟校正（SDC）方法为包含守恒律的非线性演化方程提供了一种构建任意高阶时间积分方法的系统化途径。这些方法收敛于$A$-稳定乃至$L$-稳定的配置方法，但其自身往往缺乏足够的鲁棒性。本文受Lax-Wendroff方法隐式表述的启发，发展了一类新型SDC方法。相较于全隐式方法，该系列方法仅需求解正定或半定线性系统。数值实验表明：采用Radau配置点的半隐式SDC方法在11阶以下具有$L$-稳定性，在13阶和15阶时仅需极小的耗散项。通过一维守恒问题（包括对流-扩散方程、Burgers方程、Euler方程和Navier-Stokes方程）的数值实验，验证了该方法优异的稳定性和精度。