In many applications, the governing PDE to be solved numerically contains a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is often preferred. On the other hand, if the stiff component is nonlinear, the complexity and cost per step of using an implicit method is heightened, and explicit methods may be preferred for their simplicity and ease of implementation. In this article, we analyze new and existing linearly stabilized schemes for the purpose of integrating stiff nonlinear PDEs in time. These schemes compute the nonlinear term explicitly and, at the cost of solving a linear system with a matrix that is fixed throughout, are unconditionally stable, thus combining the advantages of explicit and implicit methods. Applications are presented to illustrate the use of these methods.
翻译:在许多应用中,要以数字方式解决的管辖的PDE包含一个硬性组成部分。当该组成部分为线性时,往往倾向于一种不附带稳定性限制的隐含时间踏步方法。另一方面,如果硬性组成部分为非线性部分,则使用隐性方法的每个步骤的复杂性和成本就会提高,为了简便和便于实施,可能更倾向于采用明确的方法。在本条中,我们分析了新的和现有的线性稳定计划,以便及时将硬性非线性PDE纳入其中。这些计划明确计算非线性术语,并且以解决一条线性系统的费用为代价,用一个全套固定的矩阵来解决线性系统,是无条件稳定的,从而将明确和隐性方法的优点结合起来。提出了各种应用来说明这些方法的使用情况。