Qiu Yuchi, Chen Weitao, Nie Qing
Department of Mathematics, University of California, Irvine Irvine, CA 92697, USA.
Department of Mathematics, University of California, Riverside Riverside, CA 92507, USA.
Discrete Continuous Dyn Syst Ser B. 2019 Dec;24(12):6387-6417. doi: 10.3934/dcdsb.2019144.
The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction-diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order hybrid IIF-ETD method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction-diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method.
二阶隐式积分因子法（IIF2）由于其良好的稳定性条件，在求解刚性反应扩散方程方面很有效。此前，IIF主要应用于反应项中不包含显式时间相关项且边界条件为齐次的系统。如果将其应用于具有显式时间相关反应项的系统，我们发现IIF2需要极小的时间步长（相对于空间网格尺寸的平方）才能达到其理论二阶时间精度。尽管二阶隐式指数时间差分法（iETD2）能够准确处理显式时间相关反应，但它的计算成本比IIF2更高。在本文中，我们开发了一种混合方法，结合了这两种方法的优点，将IIF2应用于非显式时间相关的反应项，将iETD2应用于显式时间相关的反应项。二阶混合IIF-ETD方法（hIFE2）继承了IIF2较低的复杂度以及iETD2在大时间步长下保持二阶时间精度的能力。此外，它还继承了IIF2和iETD2方法处理反应扩散系统刚性时的无条件稳定性。通过一种变换，hIFE2能够准确且高效地处理非齐次边界条件。另外，这种方法可以自然地与更高空间维度系统的IIF和ETD的紧凑和阵列表示相结合。给出了包含线性和非线性反应的各种数值模拟，以证明新的hIFE方法具有卓越的稳定性、准确性和效率。
