一个浅水波方程的Semi—Lagrangian—半隐式格式试验方案

A SEMI-LAGRANGIAN AND SEMI-IMPLICIT NUMERICAL INTEGRATION SCHEME FOR THE SHALLOW WATER EQUATION

在正形投影坐标系下,半拉格朗日法与半隐式法相结合积分浅水波方程导出的模式,用一小时的时间步长,是无条件稳定。我们作了一个60小时的实例时间积分,计算稳定。半隐式积分格式导致求解一个赫姆霍茨方程,为比较其高低精度的优劣,我们提出在空间上用二阶精度和四阶精度解赫姆霍茨方程的超松驰法,同时,作了有无地形的试验。应该注意列,半拉格朗日法含有大量的插值,因而,在每一步时间步长里此积分格式比通常的欧拉法含有更多的计算,但是每一时间步长的计算量的增加却从使用很大的时间步长得到了更多的补偿。

A Semi-lagrangian algorithm associated with the semi-implicit method in the integration of the shallow water equation in a conformal conic projection is presented. The resulting model, which can be integrated with a time step of one hour, is unconditionally stable. A 60 hour real-time run of the scheme which is computationally stable is performed. The implicit integration scheme results in a Helmholtz equation. To compare the advantage and the shortcoming of higher order accuracy with those of lower order accuracy, we present a super-relaxation technique to solve the Helmholtz equation with second order and fourth order accuracy in space. Both topography and nontopography experiments are also performed. It should be noted that semi-Lagrangian method requires a great deal of interpolation. Although the scheme involves more calculations than the regular Eulerian method, the increase in the number of computations per time step more than compensated by the economy arising from the large step.