High order integration factor methods for systems with inhomogeneous boundary conditions
High order integration factor methods for systems with inhomogeneous boundary conditions
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DOI：
10.1016/j.cam.2018.08.036
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发表时间：
2019-03
期刊：
影响因子：
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通讯作者：
Sameed Ahmed;Xinfeng Liu
中科院分区：
文献类型：
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作者：
Sameed Ahmed;Xinfeng Liu
由于高阶空间导数和刚性反应，在开发求解高阶偏微分方程的数值方法时，通常需要对时间步长进行严格的时间稳定性约束。隐式积分法（IIF）及其紧化形式（cIIF）精确地处理空间导数和隐式处理反应项，通过将反应和空间导数的处理解耦，提供了良好的稳定性和效率。IIF的一个主要挑战是存储和计算由线性微分算子产生的稀疏离散化矩阵的潜在密集指数矩阵。为了节省计算成本和存储空间，引入了IIF的紧凑表示（cIIF）。另一个挑战是找到高阶空间离散化的矩阵，特别是在边界附近。本文通过一个具有四阶精度的反应扩散方程实例，将IIF方法推广到空间导数的高阶离散化，而计算成本和存储空间与一般二阶cIIF方法相似。该方法也可以有效地应用于处理其他类型的具有齐次和非齐次边界条件的偏微分方程。直接数值仿真验证了该方法的有效性和准确性。
Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential equations. Implicit integration method (IIF) method along with its compact form (cIIF), which treats spatial derivatives exactly and reaction terms implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reaction and spatial derivatives. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. The compact representation for IIF (cIIF) was introduced to save the computational cost and storage for this purpose. Another challenge is finding the matrix of high order space discretization, especially near the boundaries. In this paper, we extend IIF method to high order discretization for spatial derivatives through an example of reaction diffusion equation with fourth order accuracy, while the computational cost and storage are similar to the general second order cIIF method. The method can also be efficiently applied to deal with other types of partial differential equations with both homogeneous and inhomogeneous boundary conditions. Direct numerical simulations demonstrate the efficiency and accuracy of the approach.