Numerical simulation of three-dimensional telegraphic equation using cubic B-spline differential quadrature method

摘要

This paper employs a differential quadrature scheme that can be used for solving linear and nonlinear partial differential equations in higher dimensions. Differential quadrature method with modified cubic B-spline basis functions is implemented to solve three-dimensional hyperbolic equations. B-spline functions are employed to discretize the space variable and their derivatives. The weighting coefficients are obtained by semi-explicit algorithm. The partial differential equation results into a system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by employing a fourth stage Runge–Kutta method. Efficiency and reliability of the method has been established with five linear test problems and one nonlinear test problem. Obtained numerical solutions are found to be better as compared to those available in the literature. Simple implementation, less complexity and computational inexpensiveness are some of the main advantages of the scheme. Further, the scheme gives approximations not only at the knots but also at all the interior points in the domain under consideration. The scheme is found to be providing convergent solutions and handles different cases. High order time discretization using SSP–RK methods guarantee stability with respect to a given norm and a proper constraint on time step. Matrix method has been used for stability analysis in space and it is found to be unconditionally stable. The scheme can be used effectively to handle higher dimensional PDEs.