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.

在许多应用中，要求解的控制PDE在数值上包含一个刚性分量。当此组件为线性时，通常首选不受稳定性限制约束的隐式时间步进方法。另一方面，如果刚性分量是非线性的，则使用隐式方法的步骤的复杂性和成本会增加，并且显式方法可能会因为其简单性和易于实现而成为首选。在本文中，我们分析了新的和现有的线性稳定方案，以便及时集成刚性非线性PDE。这些方案显式地计算非线性项，并且以求解一个始终固定矩阵的线性系统为代价，它们是无条件稳定的，因此结合了显式和隐式方法的优点。