The Study on the Nonlinear Computations of the DQ and DC Methods

TL;DR

This paper enhances the differential quadrature and cubature methods for nonlinear problems by introducing Hadamard and SJT products, simplifying formulations, and improving computational efficiency, demonstrated through weather forecasting applications.

Contribution

It introduces a Hadamard product approach and SJT product for efficient nonlinear computations in DQ and DC methods, with simplified formulations and broader applicability.

Findings

The methods are more efficient for nonlinear problems than traditional techniques.

The Hadamard product simplifies the matrix formulation of DQ and DC methods.

Spherical harmonics improve handling of nonlinear equations in weather forecasting.

Abstract

This paper points out that the differential quadrature (DQ) and differential cubature (DC) methods due to their global domain property are more efficient for nonlinear problems than the traditional numerical techniques such as finite element and finite difference methods. By introducing the Hadamard product of matrices, we obtain an explicit matrix formulation for the DQ and DC solutions of nonlinear differential and integro-differential equations. Due to its simplicity and flexibility, the present Hadamard product approach makes the DQ and DC methods much easier to be used. Many studies on the Hadamard product can be fully exploited for the DQ and DC nonlinear computations. Furthermore, we first present SJT product of matrix and vector to compute accurately and efficiently the Frechet derivative matrix in the Newton-Raphson method for the solution of the nonlinear formulations. We also propose a simple approach to simplify the DQ or DC formulations for some nonlinear differential operators and thus the computational efficiency of these methods is improved significantly. We give the matrix multiplication formulas to compute efficiently the weighting coefficient matrices of the DC method. The spherical harmonics are suggested as the test functions in the DC method to handle the nonlinear differential equations occurring in global and hemispheric weather forecasting problems. Some examples are analyzed to demonstrate the simplicity and efficiency of the presented techniques. It is emphasized that innovations presented are applicable to the nonlinear computations of the other numerical methods as well.