A comparison of boundary correction methods for Strang splitting

AbstractWe analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

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Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0798 ◽

2017 ◽

Vol 473 (2199) ◽

pp. 20160798 ◽

Cited By ~ 1

Author(s):

Wenjun Cai ◽

Huai Zhang ◽

Yushun Wang

Keyword(s):

Wave Equation ◽

Acoustic Wave ◽

Acoustic Waves ◽

Conservation Law ◽

Heterogeneous Media ◽

Numerical Dispersion ◽

Discrete Version ◽

Symplectic Method ◽

Ode System ◽

Strang Splitting

We propose a novel stable and efficient dissipation-preserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude. Through introducing the conformal multi-symplectic structure, the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation. The proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law. More specifically, two subsystems in conjunction with the original damped wave equation are derived. One is actually the conservative Hamiltonian wave equation and the other is a dissipative linear ordinary differential equation (ODE) system. Standard symplectic method is devoted to the conservative system, whereas the analytical solution is obtained for the ODE system. An explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the Strang splitting technique. Stability analysis and convergence tests are given thereafter. A benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation. Further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media.

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Modified Strang splitting for semilinear parabolic problems

JSIAM Letters ◽

10.14495/jsiaml.11.77 ◽

2019 ◽

Vol 11 (0) ◽

pp. 77-80

Author(s):

Kosuke Nakano ◽

Tomoya Kemmochi ◽

Yuto Miyatake ◽

Tomohiro Sogabe ◽

Shao-Liang Zhang

Keyword(s):

Parabolic Problems ◽

Strang Splitting ◽

Semilinear Parabolic ◽

Semilinear Parabolic Problems

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Estimating a Distribution Function at the Boundary

Austrian Journal of Statistics ◽

10.17713/ajs.v49i1.801 ◽

2020 ◽

Vol 49 (1) ◽

pp. 1-23

Author(s):

Shunpu Zhang ◽

Zhong Li ◽

Zhiying Zhang

Keyword(s):

Distribution Function ◽

Kernel Estimation ◽

Kernel Estimator ◽

Distribution Functions ◽

Practical Application ◽

Real World Applications ◽

Distribution Kernel ◽

Estimation Of Distribution ◽

Simulation Results ◽

Boundary Correction

Estimation of distribution functions has many real-world applications. We study kernel estimation of a distribution function when the density function has compact support. We show that, for densities taking value zero at the endpoints of the support, the kernel distribution estimator does not need boundary correction. Otherwise, boundary correction is necessary. In this paper, we propose a boundary distribution kernel estimator which is free of boundary problem and provides non-negative and non-decreasing distribution estimates between zero and one. Extensive simulation results show that boundary distribution kernel estimator provides better distribution estimates than the existing boundary correction methods. For practical application of the proposed methods, a data-dependent method for choosing the bandwidth is also proposed.

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Emergent behaviors of discrete Lohe aggregation flows

Discrete and Continuous Dynamical Systems - Series B ◽

10.3934/dcdsb.2021308 ◽

2022 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Hyungjun Choi ◽

Seung-Yeal Ha ◽

Hansol Park

Keyword(s):

Matrix Model ◽

Unit Sphere ◽

Splitting Method ◽

Discrete Models ◽

Asymptotic State ◽

Sphere Model ◽

State Aggregation ◽

Strang Splitting ◽

Complete State ◽

Emergent Behaviors

<p style='text-indent:20px;'>The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.</p>

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Wave Propagation Across Acoustic/Biot’s Media: A Finite-Difference Method

Communications in Computational Physics ◽

10.4208/cicp.140911.050412a ◽

2013 ◽

Vol 13 (4) ◽

pp. 985-1012 ◽

Cited By ~ 13

Author(s):

Guillaume Chiavassa ◽

Bruno Lombard

Keyword(s):

Wave Propagation ◽

Numerical Methods ◽

Heterogeneous Media ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Compressional Wave ◽

Interface Conditions ◽

Immersed Interface Method ◽

Fluid Medium ◽

Strang Splitting

AbstractNumerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media. Wave propagation is described by the usual acoustics equations (in the fluid medium) and by the low-frequency Biot’s equations (in the porous medium). Interface conditions are introduced to model various hydraulic contacts between the two media: open pores, sealed pores, and imperfect pores. Well-posedness of the initial-boundary value problem is proven. Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context: a fourth-order ADER scheme with Strang splitting for time- marching; a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory; and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution. Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions, demonstrating the accuracy of the approach.

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