Composite scheme using localized relaxation with non-standard finite difference method for hyperbolic conservation laws

Kumar, Vivek; Rao, S. V. Raghurama

doi:10.1016/j.jsv.2007.09.055

Cited by 13 publications

(8 citation statements)

References 8 publications

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“…For this test problem, consider the linear advection equation as described in Equation (52), with the initial condition 29,30 defined on the interval [− 1, 1], written as:

u (x, t = 0) = \{\begin{array}{l} 9 false/ 5, & if prefix- 1 false/ 2 \leq x < 0 \\ 11 false/ 10, & if 0 \leq x < 1 false/ 2 \\ 1 false/ 2, & if 1 false/ 2 \leq x < 4 false/ 5 \\ 0, & else . \end{array}

In this test case, the computation is performed up to t = 2 with time step

Δ t = 0.006

for various values of N . Figure 8 presents the comparison of solutions obtained by the classical scheme and the proposed scheme.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A new reconstruction of numerical fluxes for conservation laws using fuzzy operators

Lochab

Kumar

2021

Numerical Methods in Fluids

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This article develops a new hybrid flux‐limited scheme for a numerical solution of the hyperbolic conservation laws by applying fuzzy logic‐based operator functions. The construction of the proposed flux‐limiter is explored using a fuzzy modifier function, having a suitable intensity. The purpose of this article is to present an efficient finite volume flux‐limited technique, derived from an entirely different subject of fuzzy mathematics, for tackling hyperbolic partial differential equations. Several standard test cases in one and two dimensions are solved numerically for demonstrating the robustness of the proposed new hybrid flux‐limited scheme.

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“…For this test problem, consider the linear advection equation as described in Equation (52), with the initial condition 29,30 defined on the interval [− 1, 1], written as:

u (x, t = 0) = \{\begin{array}{l} 9 false/ 5, & if prefix- 1 false/ 2 \leq x < 0 \\ 11 false/ 10, & if 0 \leq x < 1 false/ 2 \\ 1 false/ 2, & if 1 false/ 2 \leq x < 4 false/ 5 \\ 0, & else . \end{array}

In this test case, the computation is performed up to t = 2 with time step

Δ t = 0.006

for various values of N . Figure 8 presents the comparison of solutions obtained by the classical scheme and the proposed scheme.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A new reconstruction of numerical fluxes for conservation laws using fuzzy operators

Lochab

Kumar

2021

Numerical Methods in Fluids

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“…In it jumps at x = − 1 3 and x = 1 3 create a simple expansion fan and a strong steady right shock, respectively. In Figure 8, the exact solution is compared with Micken's non-conservative NSFDM (Micken-NC), conservative NSFDM (Micken-C), Roe's first-order upwind method with Harten's entropy fix (Harten), LxF, composite localized relaxation (CLR) scheme proposed in [25] and with the proposed CHRLRS. The limiter function (30) is used for CHRLRS.…”

Section: Non-linear Scalar Problemmentioning

confidence: 99%

“…The results in Figure 9 show that Micken's non-conservative NSFDM (Micken-NC) and conservative NSFDM (Micken-C) both fail to capture the expansion fan with sonic point. Roe's first-order upwind method with Harten's entropy fix (Harten), LxF and CLR scheme proposed in [25] capture shock wave and expansion fan but overly diffusive as compared with the proposed composite scheme (CHRLRS). The proposed composite scheme (CHRLRS) captures the shock at right location and expansive waves with higher accuracy.…”

Section: Non-linear Scalar Problemmentioning

confidence: 99%

“…Recently, a non-standard finite difference method (NSFDM) introduced by Micken [23] is given in conservative form for Burger-Fisher equation [24]. More recently, a composite scheme using localized relaxation with NSFDM is given in [25].…”

Section: Introductionmentioning

confidence: 99%

“…Note that function F does not change in the relaxation step as it is a function of u and f (u); therefore, F changes only in the convection step. To solve the convection step in relaxation system, we also define a parameter i locally over the three point stencil of grid as in [25], i.e.…”

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confidence: 99%

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Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

Dubey

Kadalbajoo

2008

Numerical Methods in Fluids

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In this work we present an upwind-based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non-linear problems by using the framework of relaxation system that converts a non-linear conservation law into a system of linear convection equations with a non-linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non-standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations.

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Finite element analysis of the singularly perturbed parabolic reaction‐diffusion problems with retarded argument

Khari

Kumar

2022

Numerical Methods Partial

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In this article, a parameter uniform Galerkin finite element method for solving singularly perturbed parabolic reaction diffusion problems with retarded argument is proposed. The solution of this class of problems exhibits parabolic boundary layers. The domain is discretized with a piecewise uniform mesh (Shishkin mesh) for spatial variable to capture the exponential behavior of the solution in the boundary layer region and backward‐Euler method on a equidistant mesh in the time direction. The method is proved to be unconditional stable and parameter uniform. The method is shown to be accurate of order O()N−1lnN2+Δt in maximum norm using Green's function approach. The convergence of the proposed method does not depend on the singular perturbation parameter.

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Composite scheme using localized relaxation with non-standard finite difference method for hyperbolic conservation laws

Cited by 13 publications

References 8 publications

A new reconstruction of numerical fluxes for conservation laws using fuzzy operators

A new reconstruction of numerical fluxes for conservation laws using fuzzy operators

Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

Finite element analysis of the singularly perturbed parabolic reaction‐diffusion problems with retarded argument

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