一类浅水波模型的数值方法

Numerical Methods for a Class of Shallow Water Equation

考虑BBM型非线性水波方程的数值方法.本文构造了二种半隐的数值格式.以BBM方程为例,严格分析了二种格式的稳定性与误差估计,证明了二种格式都是无条件稳定的.误差估计显示,线性Euler时间离散加谱Galerkin空间离散的收敛阶是O(Δt+N1-m),线性Crank-Nicolson时间离散加谱Galerkin空间离散的收敛阶是O(Δt2+N1-m).最后我们用数值例子讨论这两类方程解的长时间衰减率,并讨论扩散项、色散项、非线性项对解的衰减率的影响.数值例子表明,这两类浅水波方程的衰减率是:L2范接近-1/4;L∞范接近-1/2;H1半范接近-3/4,这与已知的理论结果是吻合的.

We turn to study the numerical solution of the shallow water equation. We propose two different schemes to numerically solve this equation. A detailed analysis is carried out for these schemes,and we prove that the overall schemes are unconditionally stable. The error estimation shows that the linearized Euler schema in time plus Fourier spectral method in space is convergent with the convergence order O（ Δt＋N^1-m）,and higher order convergences can be obtained if the second order backward differentiation or Crank-Nicolson schema are used to discretize the equation in time. At last,we use the proposed methods to investigate the asymptotical decay rate of the solutions to the shallow water wave equation. We equally discuss the role of the diffusion terms,the geometric dispersion and the nonlinearity respectively. The performed numerical experiment confirms that the decay rates in L^2-norm,L^∞-norm,and H^1-seminorm are very close to-1/4,-1/2,and-3/4 respectively. These numerical results are consistent with the known theoretical prediction.