二维声波方程的Crank-Nicolson无条件稳定方法研究

摘要

一阶速度-压力声波方程的有限差分数值模拟中,由于受到Courant-Friedrich-Levy(CFL)稳定性条件的限制,在分析精细结构问题时往往效率低下.将Crank-Nicolson(CN)方法引入到声波方程的有限差分模拟中,给出了声波方程的CN差分格式.通过Von Neuman方法推导分析了CN方法的稳定性条件,证明了该方法的无条件稳定性.同时,采用非均匀网格技术进行网格剖分,进一步提高了仿真效率,减少了内存消耗.仿真实验中,建立了二维多层精细结构的声传播模型,通过与传统时域有限差分的仿真结果进行对比分析,验证了该方法的有效性.%Considering the constraint of Courant-Friedrich-Levy (CFL) stability condition,it is time-consuming to solve the one-order velocity-pressure acoustic wave equation with the conventional finite-difference time domain (FDTD) method,especially,to analyze fine structure problems.Here,Crank-Nicolson (CN) method was introduced in finite difference simulation,the acoustic wave equation's CN difference scheme was obtained.Based on Von Neuman method,the unconditional stability condition of CN method for acoustic wave equations was derived.With the proposed method,the time step was not restricted by the CFL stability condition any more.Meanwhile,the non-uniform grid technology was used to generate mesh grids to further save internal memory and improve simulation efficiency.In simulation tests,a 2-dimentional multi-layer fine structure's sound propagation model was established.Through comparing the simulation test results with those using the traditional FDTD method,the effectiveness of the proposed method was verified.