Stormer-Numerov HDG Methods for Acoustic Waves

Cockburn, Bernardo; Fu, Zhixing; Hungria, Allan; Ji, Liangyue; Sánchez, Manuel; Sayas, Francisco–Javier

doi:10.1007/s10915-017-0547-z

Cited by 32 publications

(30 citation statements)

References 23 publications

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“…To be more specific, we present a semi-discrete HDG+ method for transient elastic waves that has a uniform-in-time optimal convergence. We show that the proof for optimal convergence can be easily obtained by using the new projection and some existing techniques in traditional HDG methods for evolutionary equations [6]. Moreover, we recover the error estimates for steady-state elasticity [19] and frequency domain elastodynamics [15] by using the projection-based analysis, and we show that the analysis can be simplified in both cases.…”

Section: Introductionmentioning

confidence: 82%

“…xp u h ptq, µy Γ " xgptq, µy Γ , (7.2d) for all θ P V h , w P W h and µ P M h . For the initial conditions of the method, we use ideas from [6]. The initial velocity 9 u h p0q is defined by using the projection in Theorem 2.1 pˆ, 9 u h p0qq " ΠpA´1εpv 0 q, v 0 ; τ q, (7.3) and the initial displacement u h p0q is defined to be the solution of the HDG+ discretization of the steady-state system…”

Section: Transient Elastodynamicsmentioning

confidence: 99%

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New analytical tools for HDG in elasticity, with applications to elastodynamics

Sayas

2019

Math. Comp.

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We present some new analytical tools for the error analysis of hybridizable discontinuous Galerkin (HDG) method for linear elasticity. These tools allow us to analyze more variants of HDG method using the projection-based approach, which renders the error analysis simple and concise. The key result is a tailored projection for the Lehrenfeld-Schöberl type HDG (HDG+ for simplicity) methods. By using the projection we recover the error estimates of HDG+ for steady-state and time-harmonic elasticity in a simpler analysis. We also present a semi-discrete (in space) HDG+ method for transient elastic waves and prove it is uniformly-in-time optimal convergent by using the projection-based error analysis. Numerical experiments supporting our analysis are presented at the end.

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Section: Introductionmentioning

confidence: 82%

Section: Transient Elastodynamicsmentioning

confidence: 99%

New analytical tools for HDG in elasticity, with applications to elastodynamics

Sayas

2019

Math. Comp.

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“…The implicit, fourth-order Störmer-Numerov method, which has been applied to acoustic and elastic waves [98][99][100], is an example of a symplectic method [101]. The Störmer-Numerov approximation for system (48) is the following:…”

Section: Runge-kutta and Symplectic Methodsmentioning

confidence: 99%

Numerical modeling of mechanical wave propagation

Seriani

Oliveira

2020

Riv. Nuovo Cim.

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The numerical modeling of mechanical waves is currently a fundamental tool for the study and investigation of their propagation in media with heterogeneous physical properties and/or complex geometry, as, in these cases, analytical methods are usually not applicable. These techniques are used in geophysics (geophysical interpretation, subsoil imaging, development of new methods of exploration), seismology (study of earthquakes, regional and global seismology, accurate calculation of synthetic seismograms), in the development of new methods for ultrasonic diagnostics in materials science (non-destructive methods) and medicine (acoustic tomography). In this paper we present a review of numerical methods that have been developed and are currently used. In particular we review the key concepts and pioneering ideas behind finite-difference methods, pseudospectral methods, finite-volume methods, Galerkin continuous and discontinuous finite-element methods (classical or based on spectral interpolation), and still others such as physics-compatible, and multiscale methods. We focus on their formulations in time domain along with the main temporal discretization schemes. We present the theory and implementation for some of these methods. Moreover, their computational characteristics are evaluated in order to aid the choice of the method for each practical situation.

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“…Therefore, the HDG methods have a significantly smaller number of globally coupled degrees of freedom comparing to DG methods. The HDG methods have been extended to many models [4,5,7,25,26]. We emphasize that the HDG method in this work is considered to be a superconvergent method.…”

Section: The Ensemble Hdg Formulationmentioning

confidence: 99%

A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs

Yu¹

2021

NMTMA

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A new second order time stepping ensemble hybridizable discontinuous Galerkin method for parameterized convection diffusion PDEs with various initial and boundary conditions, body forces, and time depending coefficients is developed. For ensemble solutions in L ∞ (0, T ; L 2 (Ω)), a superconvergent rate with respect to the freedom degree of the globally coupled unknowns for all the polynomials of degree k ≥ 0 is established. The results of numerical experiments are consistent with the theoretical findings.

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Stormer-Numerov HDG Methods for Acoustic Waves

Cited by 32 publications

References 23 publications

New analytical tools for HDG in elasticity, with applications to elastodynamics

New analytical tools for HDG in elasticity, with applications to elastodynamics

Numerical modeling of mechanical wave propagation

A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs

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