The ${l}^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials

Wen, Xin; Jin, Shi

doi:10.1137/070700681

Cited by 11 publications

(28 citation statements)

References 27 publications

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“…The convergence rate is nearly first order, which agrees with the discussion for interface problem in [13]. This is more accurate than the results in [12], where only halfth order was obtained [27]. We output the numerical solutions of the density ρ(x, y, t) with different meshes against the exact density in Figure 5.…”

Section: Example 3 Consider the 1d Liouville Equation On The Computasupporting

confidence: 70%

A Hybrid Phase Flow Method for Solving the Liouville Equation in a Bounded Domain

Wu¹,

Yang²

2011

SIAM J. Numer. Anal.

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The phase flow method was originally introduced in [28] which can efficiently compute the autonomous ordinary differential equations. In [13], it was generalized to solve the Hamiltonian system where the Hamiltonian contains discontinuous functions, for example discontinuous potential or wave speed. However, both these works require the flow map constructed on an invariant manifold. This could lead to an expensive computational cost when the invariant domain is big or even unbounded.In this paper, following the idea of [13], we propose a hybrid phaseflow method for solving the Liouville equation in the bounded domain where the flow map sits in the variant manifold of the traditional phase flow map. Moreover, with the help of some proper boundary conditions, this hybrid phase flow method could help reduce the numerical difficulty when the invariant manifold of the phase flow given by the Liouville equation is big or unbounded. Analysis of the numerical stability and convergence is given for the Liouville equation with the inflow boundary condition. We also verify the accuracy and efficiency of this algorithm by several examples related to the semiclassical limit of the Schrödinger equation.

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Section: Example 3 Consider the 1d Liouville Equation On The Computasupporting

confidence: 70%

A Hybrid Phase Flow Method for Solving the Liouville Equation in a Bounded Domain

Wu¹,

Yang²

2011

SIAM J. Numer. Anal.

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“…Note the discontinuous Hamiltonian solver Θ ∆t is second order convergence here, see Table 2, the reason for the first order convergence comes from the criteria (32)-(34). However, this is still more accurate than the finite difference and finite volume method in [14] where only halfth order was achieved [25].…”

Section: Example 1 Consider the 1d Liouville Equationmentioning

confidence: 99%

A Hybrid Phase-Flow Method for Hamiltonian Systems with Discontinuous Hamiltonians

Jin¹,

Huang³

2009

SIAM J. Sci. Comput.

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In this paper, we propose a new phase flow method for Hamiltonian systems with discontinuous Hamiltonians. In the original phase-flow method introduced by Ying and Candès [26], the phase map should be smooth to ensure the accuracy of the interpolation. Such an interpolation is inaccurate if the phase map is nonsmooth, for example, when the Hamiltonian is discontinuous. We modify the phase flow method using a discontinuous Hamiltonian solver, and establish the stability (for piecewise constant potentials) of such a solver. This extends the applicability of the highly efficient phase flow method to singular Hamiltonian systems, with a mild increase of algorithm complexity. Such a particle method can be useful for the computation of high frequency waves through interfaces.

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“…This approach can not only derive the optimal convergence rate but also give explicit coefficients in the error bound estimates. This is the first step toward establishing a convergence theory for the Hamiltonian preserving schemes for the Liouville equation with singular-both discontinuous and measure-valuedcoefficients [10][11][12].Such schemes have important applications in computational high frequency waves through heterogeneous media [7-9, 13, 14, 33]But so far only stability results are available [33]. This paper is organized as follows.…”

Section: 4)mentioning

confidence: 99%

Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients II: Some Related Binomial Coefficients Inequalities

Wen¹

2009

JCM

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We study the L 1 -error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L 1 -norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L 1 -error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].Mathematics subject classification: 65M06, 65M12, 65M25, 35F10.

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The ${l}^1$-Error Estimates for a Hamiltonian-Preserving Scheme for the Liouville Equation with Piecewise Constant Potentials

Cited by 11 publications

References 27 publications

A Hybrid Phase Flow Method for Solving the Liouville Equation in a Bounded Domain

A Hybrid Phase Flow Method for Solving the Liouville Equation in a Bounded Domain

A Hybrid Phase-Flow Method for Hamiltonian Systems with Discontinuous Hamiltonians

Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients II: Some Related Binomial Coefficients Inequalities

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