Numerical simulation of two dimensional quasilinear hyperbolic equations by polynomial differential quadrature method

Kumar, Vikas; Jiwari, Ram; Gupta, Rajesh Kumar

doi:10.1108/ec-02-2012-0030

Cited by 20 publications

(20 citation statements)

References 34 publications

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“…(ii) CTBS DQ algorithm proposed in [33] has extended for 2D problems in different forms, and it has concluded the algorithm worked nicely for the same problems. (iii) e developed algorithm is better than the DQ algorithms proposed in [31,32,34] due to more smoothness of CTBS functions.…”

Section: Discussionmentioning

confidence: 94%

“…Bellman et al [28] introduced two approaches to calculate WCs. Furthermore, to modify Bellman's approaches for finding WCs, many efforts have been carried out such as Lagrange interpolated cosine functions, spline functions, Legendre polynomials, Lagrange interpolation polynomials, and radial basis functions (see [19,[29][30][31][32][33][34][35] and the references therein) to determine these coefficients. In this study, we determine WCs with the use of CTBS functions after some modifications.…”

Section: Differential Quadrature Methodsmentioning

confidence: 99%

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A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

Kumar¹,

Singh

Köksal

2021

Journal of Mathematics

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In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

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Section: Discussionmentioning

confidence: 94%

Section: Differential Quadrature Methodsmentioning

confidence: 99%

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

Kumar¹,

Singh

Köksal

2021

Journal of Mathematics

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“…We notice that for 1 r = the truncation error vanishes completely. As , 0 h k → (with r constant), , 0 i j T → , so the difference scheme ( 13) is consistent with the wave Equation (2). The CTCS method is also stable for 1 r ≤ , [22] hence by Laxs equivalence theorem [21] it is also convergent for 1 r ≤ .…”

Section: The Ctcs Explicit Difference Schemementioning

confidence: 99%

“…For each of the methods CTCS, Crank-Nicolson and ω we use second order forward, centered and backward differences to approximate the initial derivative condition. In Section 4 we use the methods mentioned in Section 3 to solve an IBVP for (2) for different values of c (…”

Section: Introductionmentioning

confidence: 99%

A Computational Study with Finite Difference Methods for Second Order Quasilinear Hyperbolic Partial Differential Equations in Two Independent Variables

Stampolidis¹,

Gousidou-Koutita²

2018

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In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme Central Time Central Space (CTCS), Crank-Nicolson scheme, ω scheme and the method of characteristics for the numerical solution of initial and boundary value problems for the one-dimension homogeneous wave equation. The initial derivative condition is approximated by different second order difference quotients in order to examine which gives more accurate numerical results. The local truncation error, consistency and stability of the difference schemes CTCS, Crank-Nicolson and ω are also considered.

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“…The approaches for solving boundary value problems with homogeneous boundary conditions seem to be effortless in comparison to the problems with non-homogeneous boundary conditions (Kumar et al, 2013; Verma and Jiwari, 2015). The non-homogeneous boundary conditions, for which the problem has solutions, need to be homogenized with the aid of some transformations of the variable.…”

Section: Introductionmentioning

confidence: 99%

On point source influencing Love-type wave propagation in a functionally graded piezoelectric composite structure: A Green’s function approach

Singh

Das

Ray

et al. 2018

Journal of Intelligent Material Systems and Structures

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Green’s function plays an important role in solving the problems concerning point action or impulse responsible for wave motions in materials. Prime objective of the this article is to investigate the propagation behaviour of Love-type wave influenced by a point source in a composite structure comprising a functionally graded piezoelectric material layer lying over a functionally graded fibre-reinforced material half-space. Green’s function technique is adopted in order to obtain the dispersion equation, which is further reduced to the classical Love wave equation as a particular case of the problem. The effect of increasing thickness of functionally graded piezoelectric material layer on the circular frequency and wave number is unravelled and depicted graphically. Moreover, influence of heterogeneity, piezoelectricity and dielectric constant associated with functionally graded piezoelectric material layer and effect of heterogeneity parameter and corresponding magnification factor concerned with functional gradedness of functionally graded fibre-reinforced material half-space have been reported through numerical computation and graphical delineation. For sake of computation, numerical data of PZT-5H ceramics for the functionally graded piezoelectric material layer and carbon-fibre epoxy-resin for functionally graded fibre-reinforced material half-space have been considered. Comparative study is performed to elucidate the effect of presence and absence of reinforcement in functionally graded half-space on the phase velocity of Love-type wave propagating in composite structure.

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Numerical simulation of two dimensional quasilinear hyperbolic equations by polynomial differential quadrature method

Cited by 20 publications

References 34 publications

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

A Computational Study with Finite Difference Methods for Second Order Quasilinear Hyperbolic Partial Differential Equations in Two Independent Variables

On point source influencing Love-type wave propagation in a functionally graded piezoelectric composite structure: A Green’s function approach

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