不同时空格式在求解Boussinesq水波模型中的应用

Application of Different Time & Space Schemes in Boussinesq Models

为了解不同时间空间差分格式在常用的二阶或四阶Boussinesq模型中的应用,针对4组近似到O(μ2)阶完全非线性的二阶或四阶色散性的高阶Boussinesq水波方程,在非交错网格下,利用Crank-Nicolson格式、蛙跳格式、混合四阶Adams-Bashforth-Moulton格式,建立不同的数学模型。利用这些数值模型模拟波浪在潜堤上的传播变形,通过数值结果与试验结果的比较,考察时间格式及空间格式对模型的影响。结果表明:对同一方程,混合四阶Adams-Bashforth-Moulton格式和Crank-Nicolson格式均能取得较好模拟效果,蛙跳格式的模拟效果最差;二阶Boussinesq模型采用追赶法求解已能满足要求;对四阶Boussineq模型,二阶空间导数色散项亦采用四阶精度,其数值效果会更好。

To understand the application of the different time and space derivative scheme in second order or fourth order dispersive Boussinesq equations, numerical models were established in non-staggered grids based on four sets of higher order Boussinesq equations with nonlinearity accurate to O（μ^2）. In those models, time marching schemes included Crank-Nicolson scheme, Frog-loop scheme and a composite fourth order Adams-Bashforth- Mouhon scheme. Numerical simulations were done upon wave propagating over a submerged sill with these models, through the comparisons among the numerical results and the experimental data, the effects of different schemes of time and space derivative were investigated. The results shows ： For the same Boussinesq equations, the models with a composite fourth order Adams-Bashforth-Mouhon scheme or Crank-Nicloson scheme can simulate well, and those models with Frog-loop scheme simulate worse. For a second order Boussinesq model, tridiagonal matrix method is efficient. For fourth order Boussinesq model, when the second order space derivatives for sive terms adopt fourth order accuracy, the numerical results are more satisfactory. disper