Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets

Pandit, Sapna; Kumar, Manoj; Mohapatra, R. N.

doi:10.1108/hff-05-2016-0188

Cited by 11 publications

(3 citation statements)

References 34 publications

(77 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Compute the local first and second order approximation named as sparse differentiation matrices α (1) and α (2) A3. Enter the linear system (19) to get the solution at grid point A4. Solve the entered system in MATLAB via LU decomposition method and iterates upto T (final time)…”

Section: Matlab Implementation Steps Of Thementioning

confidence: 99%

“…ere are various numerical techniques available in literature for the simulation of CDM and RDM, e.g., adaptive B-spline collocation method [2], ε-uniform schemes [3], parameter uniform difference scheme [4], uniformly convergent B-spline collocation method [5], finite difference fitted schemes [6], uniformly convergent difference schemes [7,8], SDFEM [9], parameter-uniform hybrid finite difference scheme [10], finite difference domain decomposition algorithms [11], high order methods [12], Layer-adapted meshes and FEM [13], parameter uniform approximations [14], uniformly convergent scheme [15], and finite difference scheme [16]. Except these, there are many schemes for these types of problems [17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Local RBF‐FD‐Based Mesh‐free Scheme for Singularly Perturbed Convection‐Diffusion‐Reaction Models with Variable Coefficients

Jiwari

Singh

2022

Journal of Mathematics

View full text Add to dashboard Cite

This work analyze singularly perturbed convection-diffusion-reaction (CDR) models with two parameters and variable coefficients by developing a mesh-free scheme based on local radial basis function-finite difference (LRBF-FD) approximation. In the evolvement of the scheme, time derivative is discretized by forward finite difference. After that, LRBF-FD approximation is used for spatial discretization, and we obtained a system of linear equations. Then, the obtained linear system is solved by LU decomposition method in MATLAB. For numerical simulation, four singularly perturbed models are pondered to check the efficiency and chastity of the proposed scheme.

show abstract

Section: Matlab Implementation Steps Of Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local RBF‐FD‐Based Mesh‐free Scheme for Singularly Perturbed Convection‐Diffusion‐Reaction Models with Variable Coefficients

Jiwari

Singh

2022

Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…The analytical solutions for smooth, cusped and sharp waves presented in this section can be used to assess the accuracy of numerical methods for nonlinear degenerate advection-diffusion equations and, in particular, those which have been previously used to determine the solutions of the Smooth, cusped and sharp shock waves one-dimensional Burgers, modified Burgers and generalized Burgers equations, e.g. Mittal and Jiwari (2012), Korkmaz and Da g (2013), Jiwari (2015), Jiwari and Alshomrani (2017), Jiwari et al (2019) and Pandit et al (2017) and references therein, and other nonlinear wave equations, e.g. the Korteweg-de Vries, generalized regularized-long-wave, equal-width, etc., equations (Tauseef Mohyud-Din et al, 2012;Ramos, 2016;Apolinar-Fern andez and Ramos, 2018).…”

Section: Nil Upstream Boundary Conditionsmentioning

confidence: 99%

Smooth, cusped and sharp shock waves in a one-dimensional model of a microfluidic drop ensemble

Ramos

2021

HFF

View full text Add to dashboard Cite

Purpose The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model. Design/methodology/approach The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed. Findings For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest. Originality/value A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

show abstract

Wavelet strategy for flow and heat transfer in CNT-water based fluid with asymmetric variable rectangular porous channel

Pandit

Sharma

2020

Engineering with Computers

View full text Add to dashboard Cite

Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets

Cited by 11 publications

References 34 publications

Local RBF‐FD‐Based Mesh‐free Scheme for Singularly Perturbed Convection‐Diffusion‐Reaction Models with Variable Coefficients

Local RBF‐FD‐Based Mesh‐free Scheme for Singularly Perturbed Convection‐Diffusion‐Reaction Models with Variable Coefficients

Smooth, cusped and sharp shock waves in a one-dimensional model of a microfluidic drop ensemble

Wavelet strategy for flow and heat transfer in CNT-water based fluid with asymmetric variable rectangular porous channel

Contact Info

Product

Resources

About