Numerical solution of second-order two-dimensional hyperbolic equation by bi-cubic B-spline collocation method

• Published in: Mathematical sciences (Karaj, Iran) Vol. 14; no. 3; pp. 201 - 213
• Main Authors: Arora, Rajni, Singh, Swarn, Singh, Suruchi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Boundary conditions; Collocation methods; Dirichlet problem; Exact solutions; Mathematics; Mathematics and Statistics; Numerical methods; Ordinary differential equations; Original Research; Partial differential equations; SSP RK-22; 65M06; Telegraph equation; 65N06; Damped wave equation; 65M60; Collocation method; 65N30; Tri-diagonal solver
• ISSN: 2008-1359; 2251-7456
• DOI: 10.1007/s40096-020-00331-y

A method based on B-splines has been introduced for the solution of second-order nonlinear hyperbolic equation in 2-dimensions subject to appropriate initial and Dirichlet boundary conditions. We first convert the second-order equation into a system of first-order partial differential equations. Then, collocation of bi-cubic B-splines is used to discretize spatial variables and their derivatives to further obtain first-order ordinary differential equations which have block tri-diagonal structure. Computation technique is discussed to handle the thus obtained block tri-diagonal matrices, which are then solved by two-step, second-order strong-stability-preserving Runge--Kutta method (SSP RK-22). The efficiency and accuracy of the proposed method are demonstrated by its application to a few test problems and by comparing the results with analytic solutions and with the results obtained by using other numerical methods available in the literature.