A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

Haynes, Ronald D.

doi:10.4208/nmtma.2013.1130nm

“…The aim of the equidistribution principle is to concentrate the nonuniformly distributed grid points in the steep regions of the solution (see [7,13,34,40] and references therein). In this principle, the desired mapping is obtained as a solution of the nonlinear problem:…”

Section: Adaptive Nonuniform Grid Generationmentioning

confidence: 99%

Higher Order Non-Uniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

Iqbal¹,

Zegeling²

2019

Ж. вычисл. мат. и мат. физ.

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In this paper, a higher order nonuniform grid strategy is developed for solving singularly perturbed convection-diffusion-reaction problems with boundary layers. A new nonuniform grid finite difference method (FDM) based on a coordinate transformation is adopted to establish higher order accuracy. To achieve this, we study and make use of the truncation error of the discretized system to obtain a fourth-order nonuniform grid transformation. Considering a three-point central finite-difference scheme, we create not only fourth-order but even sixth-order approximations (which is the maximum order that can be obtained) by a suitable choice of the underlying nonuniform grids. Further, an adaptive nonuniform grid method based on equidistribution principle is used to demonstrate the sixth-order of convergence. Unlike several other adaptive numerical methods, our strategy uses no pre-knowledge of the location and the width of the layers. Numerical experiments for various test problems are presented to verify the theoretical aspects. We also show that other, slightly different, choices of the grid distributions already lead to a substantial degradation of the accuracy. The numerical results illustrate the effectiveness of the proposed higher order numerical strategy for nonlinear convection dominated singularly perturbed boundary value problems.

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“…The aim of the equidistribution principle is to concentrate the nonuniformly distributed grid points in the steep regions of the solution (see [7,13,34,40] and references therein). In this principle, the desired mapping is obtained as a solution of the nonlinear problem:…”

Section: Adaptive Nonuniform Grid Generationmentioning

confidence: 99%

Higher Order Nonuniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

Iqbal

¹

,

Zegeling

²

2019

Comput. Math. and Math. Phys.

0

View full text Add to dashboard Cite

In this paper, a higher order nonuniform grid strategy is developed for solving singularly perturbed convection-diffusion-reaction problems with boundary layers. A new nonuniform grid finite difference method (FDM) based on a coordinate transformation is adopted to establish higher order accuracy. To achieve this, we study and make use of the truncation error of the discretized system to obtain a fourth-order nonuniform grid transformation. Considering a three-point central finite-difference scheme, we create not only fourth-order but even sixth-order approximations (which is the maximum order that can be obtained) by a suitable choice of the underlying nonuniform grids. Further, an adaptive nonuniform grid method based on equidistribution principle is used to demonstrate the sixth-order of convergence. Unlike several other adaptive numerical methods, our strategy uses no pre-knowledge of the location and the width of the layers. Numerical experiments for various test problems are presented to verify the theoretical aspects. We also show that other, slightly different, choices of the grid distributions already lead to a substantial degradation of the accuracy. The numerical results illustrate the effectiveness of the proposed higher order numerical strategy for nonlinear convection dominated singularly perturbed boundary value problems.

show abstract

“…The aim of the equidistribution principle is to concentrate the non-uniformly distributed grid points in the steep regions of the solution (see [22,53,149,147] and references therein). In this principle, the desired mapping x(ξ) is obtained as a solution of the nonlinear problem:…”

Section: Adaptive Non-uniform Grid Generationmentioning

confidence: 99%

Numerical Methods for Nonlinear Elliptic Boundary Value Problems with Parameter Dependence

Iqbal¹

0

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Chapter 1 1.3 Nonstandard finite difference schemes The majority of differential equation models (both linear and nonlinear), which correspond to dynamical systems stemming from engineering and natural sciences, can't be solved exactly in terms of elementary functions. As a consequence, a variety of numerical methods has been constructed to find an approximate solution. One of the traditional techniques in this area, is the finite difference method [78, 11, 52, 128, 73]. Unfortunately, numerical instabilities may arise in these finite difference approximations. Normally, numerical instabilities can be recognized as • Rule 1. The order of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equations. This rule demonstrates that spurious solutions may exist, if the order of the discrete derivatives is larger than that of the derivatives that appear in the differential equations. The following example shows the consequences, if Rule 1 is violated. Consider the decay differential equation: du dt = −u. (1.5) A central finite difference approximation for ODE (1.5) gives: u n+1 + 2∆t u n − u n−1 = 0. It has the general solution [89]: u n = C 1 (r +) n + C 2 (r −) n , with r ± = −∆t ± √ 1 + (∆t) 2 , (1.6) where C 1 and C 2 are arbitrary constants. The r ± are functions of ∆t and the following result holds for large ∆t : 1.4 Singularly perturbed boundary value problems with boundary layers It is well-known that many phenomena in fluid dynamics, biology, engineering, physics and many more can be described by boundary value problems (BVPs) associated with different types ∆x j+1 − ∆x j ∆x j ∆x j+1 , c = ∆x j ∆x j+1 (∆x j + ∆x j+1). Figure 1.5: Sequence of grids with spacing h (fine grid), 2h, 4h (coarsest grid) of the set of grid points Ω h , Ω 2h and Ω 4h , respectively. direct solution: = () − No Post-smooth: relax 2 times on fine grid = MGV (* , ℎ , 2) if coarsest grid Figure 1.6: Multigrid V-cycle flow diagram This chapter is based on the article published as [148]:

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A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

Cited by 7 publications

References 35 publications

Higher Order Non-Uniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

Higher Order Non-Uniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

Higher Order Nonuniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

Numerical Methods for Nonlinear Elliptic Boundary Value Problems with Parameter Dependence

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