Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability

Eriksson, Sofia; Nordström, Jan

doi:10.1007/s10208-016-9310-3

Cited by 13 publications

(16 citation statements)

References 48 publications

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“…To simulate seismic wave propagation in the curve domain a free-surface is set at the top boundary. Characteristic boundary conditions, a first-order accurate non-reflecting boundary condition [71], are imposed at the other boundary surfaces.…”

Section: Well-posed Boundary Conditionsmentioning

confidence: 99%

Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

2020

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In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficient complex frequency-shifted (CFS) unsplit multi-axis perfectly matched layer (MPML) is implemented using the auxiliary differential equation (ADE) that is used to build higher-order time schemes for elastodynamics in the unbounded curve domain. It is derived to be compatible with SMF. The SMF framework has a general form of a hyperbolic partial differential equation (PDE) that can be expanded to different dimensions (2D, 3D) or different wave modal (SH, P-SV) without requiring significant modifications owing to a simplified process of derivation and programming. Subsequently, an energy analysis on the framework combined with initial boundary value problems is conducted, and the stability analysis can be extended to a semi-discrete approximation similarly. Thus, we propose a semi-discrete approximation based on ADE CFS-MPML in which the curve domain is discretized using the upwind summation-by-parts (SBP) operators, and where the boundary conditions are enforced weakly using the simultaneous approximation terms (SAT). The proposed method’s robustness and adequacy are illustrated by conducting several numerical simulations.

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Section: Well-posed Boundary Conditionsmentioning

confidence: 99%

Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

2020

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“…For the semi-discrete or fully discrete finite element method of the non-linear hyperbolic equation with only x or h (x, u) ≡ 1 in h (x, u), there are some research results [12,13]. If u is included in h (x, u), the error estimation will suffer or fail to reach the convergence order [14] when defining the non-linear or predictor-corrector scheme, and the error equation cannot be obtained by direct weighting method. In this paper, the finite element scheme of second-order nonsexual hyperbolic equation [15] is defined when h contains u.…”

Section: Question Descriptionmentioning

confidence: 99%

Fully discrete convergence analysis of non-linear hyperbolic equations based on finite element analysis

Zhang

2019

Applied Mathematics and Nonlinear Sciences

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With the development of modern partial differential equation (PDE) theory, the theory of linear PDE is becoming more and more perfect, . Non-linear PDE has become a research hotspot of many mathematicians. In fact, when describing practical physical problems with PDEs, non-linear problems tend to be more general than linear problems, which are close to real problems and have practical physical significance. Hyperbolic PDEs are a kind of important PDEs describing the phenomena of vibration or wave motion. The solution of hyperbolic PDE can be decomposed into the form of multiplication of vibration and vibration or of exponential function and exponential function. Generally, the energy is infinite. A full discrete convergence analysis method for non-linear hyperbolic equation based on finite element analysis is proposed. Taking second-order and fourth-order non-linear hyperbolic equation as examples, the full discrete convergence of non-linear hyperbolic equation is analysed by finite element method and the super-convergence results are obtained.

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“…The solution to (4.7) can now be written as a linear combination of eigenvectors to M , 10) where the σ j 's are determined by the boundary conditions. In (4.10), Ψ = [ψ 1 , ..., ψ n ] is the matrix of eigenvectors,…”

Section: Nrbc's For Systems Of Hyperbolic Equationsmentioning

confidence: 99%

“…Unfortunately, constructing NRBC's for a general problem in more than one space dimension is not an easy task, and one often have to resort to approximations of various parameters of the incoming waves when constructing them. This subject is treated in the classical paper by Engquist and Majda [9] and in [25,14,11,28,10]. Another way to construct NRBC's is to add buffer zones, where the incoming waves are eliminated.…”

Section: Introductionmentioning

confidence: 99%

“…As a second contribution of this thesis, we show how to use the SBP-SAT technique in time [20,23,24] to construct exact NRBC's. Our technique is performed in timedomain and it is based on the analysis performed in [9,25,10]. It is shown that the derived NRBC's result in a well-posed problem and that the corresponding numerical scheme is stable and accurate.…”

Section: Introductionmentioning

confidence: 99%

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High-order finite difference approximations for hyperbolic problems : multiple penalties and non-reflecting boundary conditions

Frenander¹

2017

Linköping Studies in Science and Technology. Dissertations

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In this thesis, we use finite difference operators with the Summation-By-Parts property (SBP) and a weak boundary treatment, known as Simultaneous Approximation Terms (SAT), to construct high-order accurate numerical schemes. The SBP property and the SAT's makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations.For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In terms of well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT's; we call the procedure the Multiple Penalty Technique (MPT).A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time.To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis, we show how to construct NRBC's by using SBP operators in time.In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal. Sammanfattning på svenskaMånga fenomen som observeras i naturen beskrivs matematiskt av partiella differentialekvationer med lämpliga rand-och initialvillkor. Dessa fenomen påträffas inom flödesdynamik, kvantmekanik och elektromagnetism, för att nämna några exempel. I allmänhet kan dessa problem inte lösas analytiskt, och måste därför modelleras med hjälp av datorer. Den spatiella domänen delas då upp i etẗ andligt antal diskreta punkter, där man söker en approximativ lösning.I den här avhandlingen används finita differensoperatorer på s.k partiell summationsform för att konstruera noggranna numeriska metoder. Med hjälp av en svag implementation av randvillkoren blir metodenäven stabil. Denna metodik ar allmän och kan tillämpas på alla problem, men i denna avhandling fokuserar vi på våg-och flödesproblem.Data måste vara tillgängligt på randen för att problemet ska va...

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Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability

Cited by 13 publications

References 48 publications

Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

Fully discrete convergence analysis of non-linear hyperbolic equations based on finite element analysis

High-order finite difference approximations for hyperbolic problems : multiple penalties and non-reflecting boundary conditions

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