We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs naturally when deriving some second order accurate schemes in time. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the second time derivative is not completely replaced by space derivatives using the PDE, but the mixed derivative is kept. If approximated in a suitable way, the resulting compact implicit scheme produces algebraic systems that have a more convenient structure than the systems derived by fully implicit schemes. We derive a high resolution TVD form of the implicit scheme for some representative hyperbolic equations in the one-dimensional case, including illustrative numerical experiments.
翻译:我们提出了一个新颖的隐含计划,用数字方法解决取决于时间的养护法问题,提出方法的核心思想是利用和估计在及时得出某些第二顺序准确计划时自然产生的解决办法的空间-时空混合衍生物,这种方法是在Lax-Wendroff(或Cauchy-Kowalevski)程序的背景下引入的,因为第二次衍生物不是完全由使用PDE的空间衍生物取代,而是保留混合衍生物。如果以适当方式加以比较,所产生的紧凑的隐含计划会产生代数系统,其结构比完全隐含计划产生的系统更方便。我们在一维中为某些具有代表性的双曲方程式提出了一种高分辨率TVD的隐含计划形式,包括说明性数字实验。