Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids

• Published in: Mathematical modelling and analysis Vol. 26; no. 1; pp. 147 - 169
• Main Authors: Ratas, Mart, Salupere, Andrus, Majak, Jüri
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 01.01.2021
• Subjects: adaptive grid; Comparative analysis; Differential equations; Exact solutions; haar wavelet method; higher order wavelet expansion; Mathematical models; Methods; Nonlinear differential equations; Nonlinear equations; nonlinear pdes; nonuniform grid; Numerical analysis; numerical simulation; Partial differential equations; Wavelet analysis; Haar wavelet method; higher order wavelet expansion; nonuniform grid; adaptive grid; numerical simulation; nonlinear PDEs
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2021.12920

The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points.