Numerical conservative solutions of the Hunter–Saxton equation

Grunert, Katrin; Nordli, Anders; Solem, Susanne

doi:10.1007/s10543-020-00835-y

Cited by 6 publications

(14 citation statements)

References 38 publications

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“…Thus, the linear and the exact soliton reconstruction coincide for the Hunter-Saxton equation. As a matter of fact, in [35] there has recently been developed a fully discrete numerical method for conservative solutions of the Hunter-Saxton equation which is primarily set in Eulerian coordinates, but employs characteristics to handle wave breaking.…”

Section: Derivation Of the Semidiscretizationmentioning

confidence: 99%

“…, n −1}. Thus, (35) provides a far more practical approach for computing the right hand side of (30) than the identities (31), especially for numerical methods, as there is no need to compute the kernels in (29). Indeed, if one uses an explicit method to integrate in time, given y, U and h we can solve (35) to obtain the corresponding R and Q.…”

Section: Presentation Of the Scheme For Global In Time Solutionsmentioning

confidence: 99%

“…We mentioned in Sect. 3 that it is computationally advantageous to solve the matrix system (35) when computing R and Q in the right-hand side of (34), or alternatively (30). Indeed, solving this nearly tridiagonal system should have a complexity close to O(n) when solved efficiently.…”

Section: Variational Finite Difference Lagrangian Scheme From Sectmentioning

confidence: 99%

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A numerical study of variational discretizations of the Camassa–Holm equation

Galtung

Grunert

2021

Bit Numer Math

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We present two semidiscretizations of the Camassa–Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa–Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.

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Section: Derivation Of the Semidiscretizationmentioning

confidence: 99%

Section: Presentation Of the Scheme For Global In Time Solutionsmentioning

confidence: 99%

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A numerical study of variational discretizations of the Camassa–Holm equation

Galtung

Grunert

2021

Bit Numer Math

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“…More recently, a Godunov-inspired scheme [25,Chp. 12.1] for conservative solutions based on tracking the solution along characteristics, but set in Eulerian coordinates, was introduced in [14]. The scheme was proved to converge and a convergence rate prior to wave breaking was derived.…”

Section: Introductionmentioning

confidence: 99%

A numerical algorithm for $α$-dissipative solutions of the Hunter--Saxton equation

Christiansen¹,

Grunert²,

Nordli³

et al. 2023

Preprint

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A convergent numerical method for α-dissipative solutions of the Hunter-Saxton equation is derived. The method is based on applying a tailormade projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step that introduces any approximation error. It is therefore crucial that its design ensures not only a good approximation of the initial data, but also that errors due to the energy dissipation at later times remain small. Furthermore, it is shown that the main quantity of interest, the wave profile, converges in L ∞ for all t ≥ 0, while a subsequence of the energy density converges weakly for almost every time.

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“…The reciprocal transformations [24] , Homotopy decomposition method [25] , bivariate generalized fractional order of the Chebyshev functions (BGFCF) [26] , cubic trigonometric B-Spline collocation method [27] , collocation method [28] , Harr wavelet quasilinearization approach [29] , and Lipschitz metric [30] , time marching scheme [31] are applied to study the diffusion of neumatic LCs. The generalized Hunter-Saxton equation is considered using integrability structures [32] , Numerical solutions of HSE using Laguerre wavelet and by using efficient approach on time domains is presented in [33] , [34] , [35] .…”

Section: Introductionmentioning

confidence: 99%

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

Ahmad

Ilyas

Kutlu

et al. 2021

Heliyon

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In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.

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Numerical conservative solutions of the Hunter–Saxton equation

Cited by 6 publications

References 38 publications

A numerical study of variational discretizations of the Camassa–Holm equation

A numerical study of variational discretizations of the Camassa–Holm equation

A numerical algorithm for $α$-dissipative solutions of the Hunter--Saxton equation

Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

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