
Unconditionally optimal convergence of an energyconserving and linearly implicit scheme for nonlinear wave equations
In this paper, we present and analyze an energyconserving and linearly ...
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A positivitypreserving secondorder BDF scheme for the CahnHilliard equation with variable interfacial parameters
We present and analyze a new secondorder finite difference scheme for t...
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Sharp L^∞ estimates of HDG methods for Poisson equation II: 3D
In [SIAM J. Numer. Anal., 59 (2), 720745], we proved quasioptimal L^∞ ...
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DataDriven Theoryguided Learning of Partial Differential Equations using SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE)
The measured spatiotemporal response of various physical processes is ut...
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Computing ground states of BoseEinstein Condensates with higher order interaction via a regularized density function formulation
We propose and analyze a new numerical method for computing the ground s...
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Variational and numerical analysis of a 𝐐tensor model for smecticA liquid crystals
We analyse an energy minimisation problem recently proposed for modellin...
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Effects of roundtonearest and stochastic rounding in the numerical solution of the heat equation in low precision
Motivated by the advent of machine learning, the last few years saw the ...
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Error estimates of energy regularization for the logarithmic Schrodinger equation
The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in different applications. Due to the singularity of the logarithmic function, it introduces tremendous difficulties in establishing mathematical theories and in designing and analyzing numerical methods for PDEs with logarithmic nonlinearity. Here we take the logarithmic Schrödinger equation (LogSE) as a prototype model. Instead of regularizing f (ρ) = ln ρ in the LogSE directly as being done in the literature , we propose an energy regularization for the LogSE by first regularizing F (ρ) = ρ ln ρ–ρ near ρ = 0 + with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrödinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter 0 < ϵ≪ 1. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically. Error estimates are also established for solving the ERLogSE by using LieTrotter splitting integrators. Numerical results are reported to confirm our error estimates of the energy regularization and of the timesplitting integrators for the ERLogSE. Finally our results suggest that energy regularization performs better than regularizing the logarithmic nonlinearity in the LogSE directly.
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