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A stiffly stable semi-discrete scheme for the damped wave equation on the half-line using SBP and SAT techniques

Thi Hoai Thuong Nguyen, Benjamin Boutin

Abstract

This paper investigates the stability of both the semi-discrete and the implicit central scheme for the linear damped wave equation on the half-line, where the spatial boundary is characteristic for the limiting equation. The proposed schemes incorporate a discrete boundary condition designed to guarantee the uniform stability of the IBVP, regardless of the stiffness of the source term or the spatial step size. Stability estimates for the semi-discrete scheme are established using the summation-by-parts (SBP) and simultaneous-approximation-term (SAT) penalty techniques, building on the continuous framework analyzed by Xin and Xu (2000).

A stiffly stable semi-discrete scheme for the damped wave equation on the half-line using SBP and SAT techniques

Abstract

This paper investigates the stability of both the semi-discrete and the implicit central scheme for the linear damped wave equation on the half-line, where the spatial boundary is characteristic for the limiting equation. The proposed schemes incorporate a discrete boundary condition designed to guarantee the uniform stability of the IBVP, regardless of the stiffness of the source term or the spatial step size. Stability estimates for the semi-discrete scheme are established using the summation-by-parts (SBP) and simultaneous-approximation-term (SAT) penalty techniques, building on the continuous framework analyzed by Xin and Xu (2000).

Paper Structure

This paper contains 12 sections, 10 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Description of the semi-discrete numerical scheme
  4. Main result
  5. The energy estimates for the linear damped wave equation
  6. Proof of Theorem \ref{['main_results']}
  7. Proof of Theorem \ref{['main_results_full']}
  8. Choice of the SAT parameters
  9. Numerical experiment
  10. Energy behaviour
  11. Convergence experiments
  12. Technical lemmas

Key Result

Theorem 1.1

Let $B_u>0$, $B_v\in\mathbb{R}$ and $(\alpha,\beta)$ be a SAT-parameter in the sense of Definition Definition4. Assume that the parameters $\Delta x\in (0,1]$ and $\varepsilon >0$ satisfy the discrete strict dissipativity condition DSDC. For any $T>0$, there exists $C_T>0$ such that for any $\left(f where the constant $C_T(\Delta x, \varepsilon)$ is independent of the data $f$ and $b$ and satisfie

Figures (3)

  • Figure 1: Evolution of the energy $E(t)$ for $\varepsilon=10^{-2}$ (left) and $\varepsilon=10^2$ (right). The legends are the parameters $\varepsilon / B_u / B_v / J$.
  • Figure 2: Space-time behaviour of $u$, $v$, $\sqrt{a}u+v$ and $\sqrt{a}u-v$ (from top to bottom). Relaxation parameter: $\varepsilon=10^{2}$, $\varepsilon=5.10^{-2}$, $\varepsilon = 10^{-3}$ (from left to right). Fixed parameters: $J=800$. $B=(1,1)^T$.
  • Figure 3: In the legends are the parameters $\varepsilon / J$. Fixed boundary parameter: $B=(1,1)^T$. Representation at time $t=0.6$ of $u_j$ and $v_j$, then for $t\in[0,0.6]$ of $E(t)$, $BU_0(t)-b(t)$ and $BU_1(t)-b(t)$ (from top to bottom). Relaxation parameters: $\varepsilon=10^{2}$, $\varepsilon=5.10^{-2}$, $\varepsilon = 10^{-3}$ (from left to right).

Theorems & Definitions (20)

  • Definition 1.1: SAT-parameter
  • Theorem 1.1: Semi-discrete scheme
  • Theorem 1.2: Implicit scheme
  • Proposition 3.1
  • proof
  • Remark
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • ...and 10 more