Structure-Preserving Combined High-Order Compact Schemes for Multiple Order Spatial Derivatives Differential Equations
Structure-Preserving Combined High-Order Compact Schemes for Multiple Order Spatial Derivatives Differential Equations
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DOI：
10.1007/s10915-023-02219-0
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发表时间：
2023-05
影响因子：
2.5
通讯作者：
Lan Wang;L. Kong;Meng Chen;Peng Fei Zhu;Huacheng Guo
中科院分区：
文献类型：
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作者：
Lan Wang;L. Kong;Meng Chen;Peng Fei Zhu;Huacheng Guo
对于具有多阶空间导数的微分方程，传统的高阶紧致离散化方法存在一些不足。其中至少有一个是由于矩阵的多重逆操作而降低了计算效率。这促使我们设计一种新的紧化方法，即组合高阶紧化方法。这种方法的基本思想是同时求解所有的空间导数。然后，将其用于求解包含一阶导数和二阶导数的耦合非线性Schrödinger （CNLS）方程。该格式不仅比同阶的标准HOC格式和标准有限差分法更紧凑、更精确，而且可以构造保结构格式。它保留了辛结构和质量，有时还保留了能量和动量。数值实验表明，该方法能较准确、有效地模拟CNLS方程。质量和动量完全保持不变。在某些特殊情况下，能量被保留了下来。
For differential equations with multiple order spatial derivatives, there are some shortcomings by the classical high order compact (HOC) discretization. At least one of them is reducing the computational efficiency due to the multiple inverse manipulation of matrices. This motivates us to design a new kind of compact method what is called combined high order compact methods. The basic idea lying in this kind of method is to solve all the spatial derivatives simultaneously. Then, it is used to solve coupled nonlinear Schrödinger (CNLS) equations which contain both the first and second order derivatives. This scheme is not only more compact and accurate than standard HOC scheme and standard finite difference method with the same order, but also it can construct structure-preserving schemes. It preserves the symplectic structure and mass, and sometimes energy and momentum. Numerical experiments indicate that the new scheme can simulate the CNLS equations very accurately and efficiently. The mass and momentum are exactly preserved. The energy is preserved in some especially cases.