A grid redistribution method for singular problems

Ditkowski, Adi; Gavish, Nir

doi:10.1016/j.jcp.2008.11.035

Cited by 10 publications

(29 citation statements)

References 20 publications

(41 reference statements)

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“…The computations of singular BNLS solutions that focus by factors of 10 8 necessitated the usage of adaptive grids. For our simulations we developed a modified version of the Static Grid Redistribution method [RW00,DG09], which is much easier to implement in the biharmonic problem, and is easily extended to other evolution equations, such as the nonlinear heat and biharmonic nonlinear heat equations [BFG09]. The method of [DG09] also includes a mechanism for the prevention of under-resolution in the non-singular region.…”

Section: Numerical Methodologymentioning

confidence: 99%

“…In order to resolve the solution at both the singular and nonsingular regions, we use an adaptive grid. We generate the adaptive grids using the Static Grid Redistribution (SGR) method, which was first introduced by Ren and Wang [RW00], and later simplified and improved by Gavish and Ditkowsky [DG09]. Using this approach, the solution is allowed to propagate (self-focus) until it becomes under-resolved.…”

Section: Numerical Method: Adaptive Mesh Constructionmentioning

confidence: 99%

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Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation

2010

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We present new singular solutions of the biharmonic nonlinear Schrödinger equationThese solutions collapse with the quasi self-similar ring profile ψQ B , whereis the ring width that vanishes at singularity, rmax(t) ∼ r0L α (t) is the ring radius, and α = 4−σThe blowup rate of these solutions is 1 3+α for 4/d ≤ σ < 4, and slightly faster than 1/4 for σ = 4. These solutions are analogous to the ring-type solutions of the nonlinear Schrödinger equation.

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Section: Numerical Methodologymentioning

confidence: 99%

Section: Numerical Method: Adaptive Mesh Constructionmentioning

confidence: 99%

Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation

2010

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“…There are several other methods that can track the blow-up dynamics. For example, one may use the adaptive mesh method in [1], moving mesh method in [4], iterative grid redistribution method in [32] for multi-dimensions and [8] for the 1d case, see also discussion on numerical treatments in [2]. These methods, unlike the dynamic rescaling method, do not need the prior knowledge of the scaling of the singular part and can deal with more general blow-up dynamics cases.…”

Section: Blow-up Dynamics For the Nls Equationmentioning

confidence: 99%

Blow-up dynamics in the mass super-critical NLS equations

Yang

Roudenko

Zhao

2019

Physica D: Nonlinear Phenomena

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“…We solve the BNLS in the case d = 1, σ = 6 with the initial condition ψ 0 (x) = 2e −x 4 . In this case, the calculated value of κ(d = 1, σ = 6) is κ = lim Figure 12: The grid-spacing ∆r m obtained using the SGR method of [10] for a peak-type singular solution of the BNLS. A) The grid generated the original method of [10] at focusing level of L = 10 −6 .…”

Section: Simulationsmentioning

confidence: 99%

“…In this case, the calculated value of κ(d = 1, σ = 6) is κ = lim Figure 12: The grid-spacing ∆r m obtained using the SGR method of [10] for a peak-type singular solution of the BNLS. A) The grid generated the original method of [10] at focusing level of L = 10 −6 . The Singular and non-singular regions are well-resolved, but the transition region ∆r m displays a discontinuity.…”

Section: Simulationsmentioning

confidence: 99%

Singular Solutions of the Biharmonic Nonlinear Schrödinger Equation

Baruch¹,

Fibich²,

Mandelbaum³

2010

SIAM J. Appl. Math.

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Abstract. We consider singular solutions of the L 2 -critical biharmonic nonlinear Schrödinger equation. We prove that the blowup rate is bounded by a quartic-root, the solution approaches a quasi-self-similar profile, and a finite amount of L 2 -norm, which is no less than the critical power, concentrates into the singularity. We also prove the existence of a ground-state solution. We use asymptotic analysis to show that the blowup rate of peak-type singular solutions is slightly faster than that of a quartic-root, and the self-similar profile is given by the ground-state standing wave. These findings are verified numerically (up to focusing levels of 10 8 ) using an adaptive grid method. We also use the spectral renormalization method to compute the ground state of the standing-wave equation, and the critical power for collapse, in one, two, and three dimensions.

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A grid redistribution method for singular problems

Cited by 10 publications

References 20 publications

Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation

Ring-type singular solutions of the biharmonic nonlinear Schrödinger equation

Blow-up dynamics in the mass super-critical NLS equations

Singular Solutions of the Biharmonic Nonlinear Schrödinger Equation

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