三维抛物方程基于特征正交分解的差分格式及其数值模拟

The finite difference method based on POD for three dimensional parabolic problems

先用有限差分格式计算出三维抛物方程瞬时解构成的数据集合,再用特征正交分解和奇值分解求出这数据集合的元素的最优正交基函数,结合Galerkin投影方法导出了三维抛物方程具有较高精度的低维模型。并给出了特征正交分解格式解和有限差分格式解的误差分析,数值例子表明特征正交分解格式解和有限差分格式解的误差与理论分析结果是一致的,从而验证了特征正交分解方法的有效性.

In this paper, the ensemble of date made up of transient solution of three dimensional parabolic equation is obtained by using finite difference scheme. Then the optimal othogonal bases are found and used to reconstruct the elements of the ensemble with POD and SVD . Combining the above approach with a Galerkin projection procedure yields a new optimizing FDS model of lower dimentions and high accuracy for three dimensional parabolic problems. The error analysis between the POD approximate solution and full FDS solution is presented. Numerical example is presented illustrating that the error between the POD approximate solution and full FDS solution is consistent with theoretical analysis results, thus validating the feasibility and efficiency of POD method.