High accuracy implicit variable mesh methods for numerical study of special types of fourth order non-linear parabolic equations

Mohanty, R. K.; Kaur, Deepti

doi:10.1016/j.amc.2015.10.036

Cited by 13 publications

(3 citation statements)

References 23 publications

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“…Table 8 shows the absolute errors obtained in problem 6. In Table 9, for different values of the parameters k, t, h and n and at different points of x, for t = 0.02 and t = 0.05 respectively, shows the maximum absolute errors obtained when the SBUI was compared with the method in [20]. This shows that the SBUI performed better.…”

Section: J41 · · · J44mentioning

confidence: 93%

A six-step Block Uniﬁcation Integrator for Numerical Solution of Fourth Order Boundary Value Problems

Modebei¹,

Adeniyi²

2018

(GLM)

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In this paper, a new 7th order continuous finite difference methods is proposed. These methods are derived using the Chebyshev polynomials as basis functions. The collocation approach is employed to obtain the main methods and additional methods used for solving general nonlinear fourth order two and four-points boundary value problems. Several numerical examples are shown to illustrate the strength of the method. To show the robustness of this method for high accuracy, we applied the method of line to discretize PDEs into system of fourth order ODEs and thus use the derived method to obtain approximate solution for the PDEs. The approximate solution obtained using the proposed methods is compared to the exact solutions of the problem, and other methods from existing literature. The Convergence of these methods is also guaranteed.

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Section: J41 · · · J44mentioning

confidence: 93%

A six-step Block Uniﬁcation Integrator for Numerical Solution of Fourth Order Boundary Value Problems

Modebei¹,

Adeniyi²

2018

(GLM)

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“…Doss and Nandini [21] give a MFE method for the extended Fisher-Kolmogorov equation. In [31], Mohanty and Kaur solved a class of fourth order non-linear parabolic equations by high accuracy implicit variable mesh methods. In [22], Wang et al solved the extended Fisher-Kolmogorov equation by using a new linearized Crank-Nicolson MFE scheme.…”

Section: Introductionmentioning

confidence: 99%

TGMFE algorithm combined with some time second-order schemes for nonlinear fourth-order reaction diffusion system

Yin

Liu

et al. 2019

Results in Applied Mathematics

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In this article, a two-grid mixed finite element (TGMFE) method with some second-order time discrete schemes is developed for numerically solving nonlinear fourth-order reaction diffusion equation. The two-grid MFE method is used to approximate spatial direction, and some second-order θ schemes formulated at time t k−θ are considered to discretize the time direction. TGMFE method covers two main steps: a nonlinear MFE system based on the space coarse grid is solved by the iterative algorithm and a coarse solution is arrived at, then a linearized MFE system with fine grid is considered and a TGMFE solution is obtained. Here, the stability and a priori error estimates in L 2 -norm for both nonlinear Galerkin MFE system and TGMFE scheme are derived. Finally, some convergence results are computed for both nonlinear Galerkin MFE system and TGMFE scheme to verify our theoretical analysis, which show that the convergence rate of the time second-order θ scheme including Crank-Nicolson scheme and second-order backward difference scheme is close to 2, and that with the comparison to the computing time of nonlinear Galerkin MFE method, the CPU-time by using TGMFE method can be saved.

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“…Ureña et al, 25 recently presented generalized finite-difference discretization using moving least squares approximation for solving nonlinear parabolic PDEs with an irregular cloud of grid points in two dimensions. Mohanty and Kaur, 27 obtained a high-order accuracy scheme to fourthorder nonlinear parabolic PDEs using variable grids and computed Boussinesq and Benjamin-Ono equations. Two-dimensions Burgers equations are numerically solved by modified B-spline (Bi-cubic) finite elements that provide approximate solutions at the grid points in the solution domain along with minimal computational work 28 .…”

mentioning

confidence: 99%

A third (fourth) order stable computational scheme for 2D Burgers type nonlinear parabolic partial differential equations on a non-uniformly spaced grid network

Jha,

Wagley

2024

Preprint

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An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burger’s type in two-dimensions. These nonlinear PDEs are essential because of the description of various mechanisms in engineering and physics. The nonlinear convective and advection are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization so that one discrete equation leads to the accuracy of orders three or four depending upon the choice of the grid network. The consistency and stability preserving third-order spatial accuracy and second-order accurate time discretization are described by Fourier analysis applied to the linearized error equations. The scheme is used for solving celebrated nonlinear PDEs, such as the non-degenerate convection-diffusion equation, generalized Burgers-Huxley equation, Buckley-Leverett equation, and Burgers-Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.

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High accuracy implicit variable mesh methods for numerical study of special types of fourth order non-linear parabolic equations

Cited by 13 publications

References 23 publications

A six-step Block Uniﬁcation Integrator for Numerical Solution of Fourth Order Boundary Value Problems

A six-step Block Uniﬁcation Integrator for Numerical Solution of Fourth Order Boundary Value Problems

TGMFE algorithm combined with some time second-order schemes for nonlinear fourth-order reaction diffusion system

A third (fourth) order stable computational scheme for 2D Burgers type nonlinear parabolic partial differential equations on a non-uniformly spaced grid network

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