四阶龙格-库塔方法的一种改进算法及地震波场模拟

提出了求解波动方程的四阶龙格-库塔方法的一种改进算法.首先将原四阶龙格-库塔方法合并为两级格式, 然后在第一级中引入加权参数以获得加权算法. 针对这种改进方法，研究了它的稳定性条件; 对一维问题导出了频散关系, 给出了数值频散结果，并与四阶的 Lax-Wendroff (LWC) 方法和位移-应力交错网格方法进行了对比; 对二维问题, 使用我们的改进方法、四阶LWC和交错网格三种方法进行了声波波场模拟, 并进行了计算效率分析和不同方法计算结果的比较; 最后选取两个层状介质模型进行了声波和弹性波波场模拟. 数值结果表明，本文的改进方法具有非常弱的数值频散和高的计算效率, 是一种在地震勘探领域具有巨大应用潜力的数值方法.

An improved algorithm of the fourth-order Runge-Kutta method and seismic wave-field simulation

In this article, we present an improved algorithm of the fourth-order Runge-Kutta (RK) method to solve the wave equations. We first change the original fourth-order Runge-Kutta method into a 2-stage scheme, and then introduce a weighting parameter in the first stage to obtain a weighted scheme. To study this new improved method, first of all, we analyze its stability condition for 1D and 2D cases. Secondly, we derive the dispersion relation for 1D problem and give the numerical dispersion results, and compare the method against the fourth-order Lax-Wendroff correction (LWC) and the displacement-stress staggered-grid methods. Thirdly, for 2D case we use the improved RK, LWC and staggered-grid methods to simulate the acoustic wave fields, and present some comparisons of the computational efficiency and numerical results for different methods. Finally, two layered-medium models are further selected to investigate the computational validity of the acoustic and elastic wave-field simulations. These numerical results show that the improved method has weak numerical dispersion, high computational efficiency, and great potentiality of application in seismic exploration.