Time Filter Method for Solving Partial Differential Equations

Partial differential equations have a wide range of applications in many areas of computational fluid dynamics. However, most of these equations cannot be solved directly. Hence it is crucial to establish numerical methods that can solve them efficiently. For numerical methods, accuracy and effectiveness are the core of an algorithm. The filtering algorithm is a numerical post-processing algorithm based on the original computational code of a complex system. The method is based on adding a simple time filter to the original numerical scheme to improve the time accuracy without additional computational complexity, and is widely used in fluid problems. In this paper, we describe the time filters, some numerical methods based on time filter, and the application of time filters to several equations. Moreover, we design the time variable filter to the Navier-Stokes equations.