Adaptive numerical method for Burgers-type nonlinear equations

Soheili, Ali Reza; Kerayechian, Asghar; Davoodi, Nooshin

doi:10.1016/j.amc.2012.09.019

Cited by 10 publications

(16 citation statements)

References 18 publications

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“…A key issue in the MMPDE method is choosing an appropriate monitor function, so that its adaptive re-meshing process effectively works. For resolving shock waves in hyperbolic conservation laws 18) , the so-called arc-type monitor functions were proposed and successfully applied to a variety of test and real cases, ranging from the inviscid Burgers-type equations 19) to the relativistic fluid dynamical equations governing evolution of galaxy 20) . Another key issue in the MMPDE method is conservation property, which is one of the most important mathematical and physical properties equipped with the conservation laws, such as the KFEs and governing equations of hydrodynamics 21) .…”

Section: (2) Mmpde Methods A) Outline Of the Mmpde Methodsmentioning

confidence: 99%

“…Assembling Eq. (19) for every i with the abovepresented numerical fluxes leads to a linear system of ordinary differential equations (ODEs) that governs temporal evolution of the nodal solutions i w , which is subject to an appropriate initial condition at the initial time in this paper. Temporal integration of the system of ODEs can be performed with an available numerical method.…”

Section: (1) Dfv Scheme With Fitting Techniquementioning

confidence: 99%

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An Adaptive Finite Volume Scheme for Kolmogorov's Forward Equations in 1-D Unbounded Domains

Yaegashi

Yoshioka

Unami

et al. 2015

J. JSCE

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This paper proposes and validates a numerical method based on the unconditionally stable dual-finite volume (DFV) scheme for Kolmogorov's forward equations (KFEs) in 1-D unbounded domains, which can be optionally equipped with a mass-conservative moving mesh partial differential equation (MMPDE) method. A KFE is a conservative and linear parabolic partial differential equation (PDE) governing spatio-temporal evolution of a probability density function (PDF) of a continuous time stochastic process. A variable transformation method is proposed for effectively solving the KFEs in 1-D bounded domains. Application of the DFV scheme to a series of test cases demonstrates its satisfactory computational accuracy, robustness, and versatility for both steady and unsteady problems. Impacts of modulating a parameter in the variable transformation method on computational performance of the DFV scheme are then numerically assessed. Advantages and disadvantages of using the MMPDE method are also investigated.

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Section: (2) Mmpde Methods A) Outline Of the Mmpde Methodsmentioning

confidence: 99%

Section: (1) Dfv Scheme With Fitting Techniquementioning

confidence: 99%

An Adaptive Finite Volume Scheme for Kolmogorov's Forward Equations in 1-D Unbounded Domains

Yaegashi

Yoshioka

Unami

et al. 2015

J. JSCE

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“…We choose these problems as our test examples because they have been used extensively in the adaptive methods literature. It is worth emphasizing that these problems are also the ones used in [6,9,29,30,33].…”

Section: Methodsmentioning

confidence: 99%

Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

Gao

Zhang

2016

Algorithms

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This paper proposes moving mesh strategies for the moving mesh methods when solving the nonlinear time dependent partial differential equations (PDEs). Firstly we analyse Huang's moving mesh PDEs (MMPDEs) and observe that, after Euler discretion they could be taken as one step of the root searching iteration methods. We improve Huang's MMPDE by adding one Lagrange speed term. The proposed moving mesh PDE could draw the mesh to equidistribution quickly and stably. The numerical algorithm for the coupled system of the original PDE and the moving mesh equation is proposed and the computational experiments are given to illustrate the validity of the new method.

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“…Budd et al in [7] applied mesh movement. Soheili et al used a moving-mesh PDE (MMPDE) approach [8]. In [9], Ramadan et al suggested a method based on collocation of septic B-splines over finite elements for numerical solutions of the non-linear Burgers equation.…”

Section: Introductionmentioning

confidence: 99%

Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

Skandari

Mohammadizadeh

et al. 2019

Symmetry

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This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

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Adaptive numerical method for Burgers-type nonlinear equations

Cited by 10 publications

References 18 publications

An Adaptive Finite Volume Scheme for Kolmogorov's Forward Equations in 1-D Unbounded Domains

An Adaptive Finite Volume Scheme for Kolmogorov's Forward Equations in 1-D Unbounded Domains

Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

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