高维波动方程数值模拟的隐式分裂有限差分格式

Implicit splitting finite difference scheme for numerical simulation of multi-dimensional wave eguation.

波动方程数值模拟的有限差分格式有隐式差分格式和显式差分格式两种,各有优点和缺点。针对高维波动方程提出了一种新的隐式分裂有限差分格式。其基本原理是:首先将高维波动方程按传播方向分解为一系列的一维波动问题,然后分别沿各方向隐式求解。该格式包含了x,y,z三个方向相互独立的一维隐式差分格式,每个方向的一维格式在数值离散后归结为一个三对角矩阵问题,可以用追赶法快速地求解。将该格式从时间一空间域变换至时间-波数域,证明此格式可以通过适当地选取参数来提高计算精度,保证计算过程的稳定性和与八阶显式差分格式同样的频散特性。脉冲响应数值计算表明,隐式分裂有限差分格式与显式差分格式相比数值频散小,频散误差接近,频散关系平滑。盐丘模型数值计算表明,隐式分裂有限差分格式与八阶显式差分格式具有同样的频散特性,但减少了计算量。

Implicit finite-difference scheme and explicit finite-difference scheme are two basic types of finite-difference schemes for numeri- cal simulation of wave equation.Each has its own advantages and disadvantages.We proposed a new implicit splitting finite-differ- ence scheme for solving the multi-dimensional wave equation.In this scheme,the multi-dimensional wave equation is firstly split in- to several independent one-dimensional equations and then solved implicitly along each direction.Each one-dimensional equation leads to a tri-diagonal matrix and can be efficiently solved by cha- sing algorithm.By transforming the finite-difference scheme into t-k domain from time-space domain,the computation accuracy can be improved by properly choosing modeling parameters,stability condition can he satisfied,and the dispersion can be reduced to that of 8^(th) order explicit finite-difference schemes.The impulse response test shows that the dispersion behavior of implicit splitting finite- difference scheme is smoother than that of explicit finite-difference schemes.Numerical simulation of a salt model indicates the new method allows bigger time step and requires less memory storage, while keeps the dispersion to the same level as that of 8^(th) order ex- plicit finite-difference.