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.

中文翻译：

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双三次B样条搭配法求解二阶二维双曲方程

引入了一种基于B样条的方法来求解二维二维非线性双曲方程，该方程在适当的初始边界条件和Dirichlet边界条件下进行求解。我们首先将二阶方程转换为一阶偏微分方程组。然后，利用双三次B样条的搭配来离散空间变量及其导数，以进一步获得具有块三对角线结构的一阶常微分方程。讨论了计算技术以处理由此获得的块三对角矩阵，然后通过两步，保持高阶二阶Runge-Kutta方法（SSP RK-22）对其进行求解。