一种新的时间依赖对流占优扩散问题的分裂方法
A New Splitting Method for Time-dependent Convection-dominated Diffusion Problems
AI总结：
提出一种将对流扩散系统分裂为纯对流和扩散子系统的显式-隐式分裂方法，通过多步技术扩大稳定区域，并推广至Navier-Stokes方程，无需调节稳定化参数。
摘要：
我们提出了一种新的时间依赖对流占优扩散问题的分裂方法。原始的对流扩散系统被分裂为两个子系统：纯对流系统和扩散系统。在每个时间步，依次求解一个对流问题和一个扩散问题。该方案具有以下优点：对流子问题显式求解，并引入多步技术以实质性地扩大稳定区域，使得所得方案类似于无条件稳定方案；扩散子问题始终是自伴且强制的，因此可以使用许多现有的最优预处理迭代求解器高效求解。随后，该方案被推广至Navier-Stokes方程，其中非线性对流在对流步骤中通过线性显式多步方案求解，而在扩散步骤中仅需求解一个广义Stokes问题，且所得刚度矩阵在时间推进过程中保持不变。新方案均无需针对对流占优扩散问题调节稳定化参数。数值模拟展示了新方案的单步和多步变体的稳定性、收敛性和性能。
英文摘要：
We present a new splitting method for time-dependent convection-dominated diffusion problems. The original convection diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. The scheme has the following nice features: the convection subproblem is solved explicitly and a multistep technique is introduced to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; the diffusion subproblem is always self-adjoint and coercive so that it can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme is then extended for Navier-Stokes equations, where the nonlinear convection is resolved by a linear explicit multistep scheme at the convection step, and only a generalized Stokes problem is needed to solve at the diffusion step with the resulting stiffness matrix being invariant in the time marching process. The new schemes are all free from tuning some stabilization parameters for the convection-dominated diffusion problems. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.
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