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

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...

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Published inMathematical modelling and analysis Vol. 26; no. 1; pp. 147 - 169
Main Authors Ratas, Mart, Salupere, Andrus, Majak, Jüri
Format Journal Article
LanguageEnglish
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
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Abstract 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.
AbstractList The higher order Haar wavelet method (HOHWM) is used with a nonuni-form 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. Keywords: numerical simulation, Haar wavelet method, higher order wavelet expansion, nonlinear PDEs, nonuniform grid, adaptive grid. AMS Subject Classification: 37M05; 65T60; 35Q51; 35G31; 65M50.
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.
Audience Academic
Author Majak, Jüri
Ratas, Mart
Salupere, Andrus
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  surname: Majak
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  organization: Department of Mechanical and Industrial Engineering, School of Engineering, Tallinn University of Technology, 19086 Tallinn, Estonia
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Keywords Haar wavelet method
higher order wavelet expansion
nonuniform grid
adaptive grid
numerical simulation
nonlinear PDEs
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Snippet The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’...
The higher order Haar wavelet method (HOHWM) is used with a nonuni-form grid to solve nonlinear partial differential equations numerically. The Burgers'...
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SubjectTerms 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
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Title Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids
URI https://journals.vilniustech.lt/index.php/MMA/article/view/12920
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Volume 26
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