10.1007/s40096-020-00331-y

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

  • Rajni Arora1,
  • Swarn Singh*2,
  • Suruchi Singh3,
  1. Department of Mathematics, University of Delhi, New Delhi, 110007, IN
  2. Department of Mathematics, Sri Venkateswara College, University of Delhi, New Delhi, 110021, IN
  3. Department of Mathematics, Aditi Mahavidyalaya, University of Delhi, Delhi, 110039, IN

Published in Issue 2020-06-07

How to Cite

Arora, R., Singh, S., & Singh, S. (2020). Numerical solution of second-order two-dimensional hyperbolic equation by bi-cubic B-spline collocation method. Mathematical Sciences, 14(3 (September 2020). https://doi.org/10.1007/s40096-020-00331-y
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Abstract

Abstract 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.

Keywords

  • Collocation method,
  • Damped wave equation,
  • SSP RK-22,
  • Telegraph equation,
  • Tri-diagonal solver

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