We present a super-time-stepping scheme for numerically solving parabolic partial differential equations with Dirichlet boundary conditions (BC). Using the general Forward Euler scheme, one can show that by taking varying step sizes there is the potential of propagating the solution forward in time by a greater amount than with uniform step sizes, while maintaining the same order of accuracy. As shown in [1] and [2], if one further requires that the scheme has the Convex Monotone Property (CMP), then there exists a scheme which results in linear, monotone stability of the solution. This monotone stability is highly desirable in many physical situations, such as thermal diffusion, where the physical system will not oscillate, but will behave monotonically. However, the schemes devised in [3], [4], [1], and [2] do not include situations that have a boundary condition [5], [6], and the inclusion of boundary conditions will henceforth be our focus. It is shown that a particular Runge-Kutta-Gegenbauer class of schemes [7] will maintain the CMP even in the presence of Dirichlet BC.

我们提出了一种超级时步方法，用于用Dirichlet边界条件（BC）数值求解抛物型偏微分方程。使用一般的正向Euler方案，可以表明，通过采用不同的步长，与保持一致的精度等级相比，有可能在时间上向前传播解决方案，而不是采用均匀步长。如[1]和[2]所示，如果进一步要求该方案具有凸单调性质（CMP），则存在一种方案，该方案可导致溶液的线性单调稳定性。在许多物理情况下（例如热扩散），此单调稳定性是非常需要的，在这些情况下，物理系统不会振荡，但会单调运行。但是，[3]，[4]，[1]，[2]和[2]不包括具有边界条件[5]，[6]的情况，因此，将边界条件的纳入作为我们的重点。结果表明，即使在Dirichlet BC存在的情况下，特定的Runge-Kutta-Gegenbauer类方案[7]仍将保持CMP。