scholarly journals A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

2013 ◽  
Vol 37 (5) ◽  
pp. 2802-2815 ◽  
Author(s):  
R.K. Mohanty ◽  
Venu Gopal
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Order Approximation ◽  
High Order ◽  
Nonlinear Wave Equations ◽  
High Order Approximation ◽  
Three Space

scholarly journals A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2957
Author(s):  
Jiong Weng ◽  
Xiaojing Liu ◽  
Youhe Zhou ◽  
Jizeng Wang
Keyword(s):  
Convergence Rate ◽  
Collocation Method ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
High Order ◽  
High Order Accuracy ◽  
Nonlinear Wave Equations ◽  
Spatial Discretization ◽  
Order Accuracy ◽  
Wavelet Approximation

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

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