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Journal ArticleDOI

Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation

04 Mar 2021-International Journal of Computer Mathematics (Informa UK Limited)-Vol. 98, Iss: 3, pp 461-479
TL;DR: Time discretization for modified Burgers equation is made by using finite difference approach along with a linearization technique and two meshless approachs are proposed based on delta-shaped basis functions-pseudo spectral method and barycentric rational interpolation method.
Abstract: In this paper, we solve modified Burgers equation numerically. Time discretization for modified Burgers equation is made by using finite difference approach along with a linearization technique. Fo...
Citations
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Journal ArticleDOI
TL;DR: An efficient localized meshless technique for approximating the viscoelastic wave model by decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain is proposed.

30 citations

Journal ArticleDOI
TL;DR: In this paper, a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs) is proposed, where the temporal part is discretized by finite difference method together with θ-weighted scheme.
Abstract: We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed $L_{\infty }$ , $L_{2}$ , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

19 citations

Journal ArticleDOI
TL;DR: In this article, a semi-discrete solution of the Sobolev equations is proposed by discretizing the spatial derivatives via the Legendre spectral element method (LSEM) and the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points.

15 citations

Journal ArticleDOI
TL;DR: In this article, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one and two-dimensional nonlinear advection-diffusion-reaction equations.
Abstract: In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and $$\theta -$$ weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), $$L_2$$ and $$L_\infty $$ error norms. The obtained results are then compared with those available in the literature.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the generalized equal width (GEW) equation is solved by a linearization technique and finite difference approach is utilized for time derivatives, which gives nodal values of numerical solution of the GEW equation for each time step.

9 citations

References
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Journal ArticleDOI
TL;DR: Matplotlib is a 2D graphics package used for Python for application development, interactive scripting, and publication-quality image generation across user interfaces and operating systems.
Abstract: Matplotlib is a 2D graphics package used for Python for application development, interactive scripting,and publication-quality image generation across user interfaces and operating systems

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TL;DR: In this article, the authors show how to improve the performance of NumPy arrays through vectorizing calculations, avoiding copying data in memory, and minimizing operation counts, which is a technique similar to the one described in this paper.
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TL;DR: In this article, an element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems, where moving least-squares interpolants are used to construct the trial and test functions for the variational principle.
Abstract: An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

5,324 citations

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TL;DR: This effort shows, NumPy performance can be improved through three techniques: vectorizing calculations, avoiding copying data in memory, and minimizing operation counts.
Abstract: In the Python world, NumPy arrays are the standard representation for numerical data. Here, we show how these arrays enable efficient implementation of numerical computations in a high-level language. Overall, three techniques are applied to improve performance: vectorizing calculations, avoiding copying data in memory, and minimizing operation counts. We first present the NumPy array structure, then show how to use it for efficient computation, and finally how to share array data with other libraries.

5,307 citations


"Two meshless methods based on pseud..." refers methods in this paper

  • ...Numerical computations have been performed in Python environment [37,58] with a desktop computer (Linux OS, NumPy version 1....

    [...]

Journal Article
TL;DR: Python is an excellent "steering" language for scientific codes written in other languages, but with additional basic tools, it transforms into a high-level language suited for scientific and engineering code that's often fastenough to be immediately useful but also flexible enough to be sped up with additional extensions.
Abstract: Python is an excellent "steering" language for scientific codes written in other languages. However, with additional basic tools, Python transforms into a high-level language suited for scientific and engineering code that's often fast enough to be immediately useful but also flexible enough to be sped up with additional extensions.

2,841 citations


"Two meshless methods based on pseud..." refers methods in this paper

  • ...Numerical computations have been performed in Python environment [37,58] with a desktop computer (Linux OS, NumPy version 1....

    [...]