Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850045 ◽  
Author(s):  
B. V. Rathish Kumar ◽  
Gopal Priyadarshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Computational Cost ◽  
Stability Estimates ◽  
Partial Differential ◽  
Wavelet Galerkin Method

We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

scholarly journals A Sixth Order Accuracy Solution to a System of Nonlinear Differential Equations with Coupled Compact Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Don Liu ◽  
Qin Chen ◽  
Yifan Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Sixth Order ◽  
Compact Schemes ◽  
Order Convergence ◽  
Order Accuracy ◽  
Partial Differential

A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates.

Barycentric Jacobi Spectral Method for Numerical Solutions of the Generalized Burgers-Huxley Equation

2017 ◽  
Vol 18 (1) ◽  
pp. 67-81 ◽  
Author(s):  
Edson Pindza ◽  
M. K. Owolabi ◽  
K.C. Patidar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Point Of View ◽  
Partial Differential ◽  
Computational Point ◽  
Exponential Time Differencing ◽  
Predictor Corrector ◽  
Numerical Approaches

AbstractNumerical solutions of nonlinear partial differential equations, such as the generalized and extended Burgers-Huxley equations which combine effects of advection, diffusion, dispersion and nonlinear transfer are considered in this paper. Such system can be divided into linear and nonlinear parts, which allow the use of two numerical approaches. Barycentric Jacobi spectral (BJS) method is employed for the spatial discretization, the resulting nonlinear system of ordinary differential equation is advanced with a fourth-order exponential time differencing predictor corrector. Comparative numerical results for the values of options are presented. The proposed method is very elegant from the computational point of view. Numerical computations for a wide variety of problems, show that the present method offers better accuracy and efficiency in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.

scholarly journals On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 2014
Author(s):  
Junjiang Lai ◽  
Hongyu Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
A Priori ◽  
Time Derivative ◽  
Stability Estimates ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Sharp Stability

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

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