基于Strang分裂和傅里叶插值方法的第五阶KdV-伯格斯-费舍尔方程的全离散方案
Fully discrete scheme for the fifth-order KdV-Burgers-Fisher equation using Strang splitting and Fourier collocation methods
AI总结：
本文提出并分析了结合Strang分裂与傅里叶插值方法的第五阶KdV-伯格斯-费舍尔方程的全离散方案，证明了在时间上二阶收敛，在空间上谱收敛。
摘要：
运算分裂是一种通过将复杂问题分解为更简单的子问题来求解非线性偏微分方程的有效技术。本文提出并分析了结合Strang分裂用于时间离散化与傅里叶插值方法用于空间离散化的第五阶Korteweg-de Vries-Burgers-Fisher方程（KBF）的全离散方案。特别是，傅里叶插值方法是所提出全离散方案的重要组成部分，并在适当正则性假设下在空间上具有谱精度。KBF方程通过结合Fisher反应项、伯格斯型扩散和更高阶KdV色散，描述了反应、耗散和色散机制的相互作用。方程被分解为一个线性算子和一个非线性算子，并在Strang分裂框架下求解子问题。在Sobolev空间$H^s$中进行收敛性分析。局部误差利用Banach空间的算子理论论证和Lie交换估计得到，而全局误差利用Lady Windermere's扇形论证得到。分析结果表明在时间上具有二阶收敛性，在空间上具有谱收敛性。数值结果证实了理论误差估计并展示了所提出全离散方案的准确性。
英文摘要：
Operator splitting is an effective technique for the numerical solution of nonlinear partial differential equations by decomposing a complex problem into simpler subproblems. In this study, we present and analyze a fully discrete scheme for the fifth-order Korteweg-de Vries-Burgers-Fisher equation (KBF) by combining Strang splitting for time discretization with the Fourier collocation method for spatial discretization. In particular, the Fourier collocation method is an essential component of the proposed fully discrete scheme and yields spectral accuracy in space under suitable regularity assumptions. The KBF equation describes the interaction of reaction, dissipative, and dispersive mechanisms by incorporating the Fisher reaction term together with Burgers-type diffusion and higher-order KdV dispersion. The equation is split into a linear operator and a nonlinear operator, and the resulting subproblems are solved within the Strang splitting framework. Convergence is analyzed in the Sobolev space $H^s$. The local error is derived using operator-theoretic arguments in Banach spaces together with Lie commutator estimates, while the global error is obtained using the Lady Windermere's fan argument. The analysis yields second-order convergence in time and spectral convergence in space. Numerical results confirm the theoretical error estimates and demonstrate the accuracy of the proposed fully discrete scheme.
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