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Using BDF schemes in the temporal integration of POD-ROM methods

Bosco García-Archilla, Alicia García-Mascaraque, Julia Novo

Abstract

In this paper we consider the numerical approximation of a semilinear reaction-diffusion model problem (PDEs) by means of reduced order methods (ROMs) based on proper orthogonal decomposition (POD). We focus on the time integration of the fully discrete reduced order model. Most of the analysis in the literature has been carried out for the implicit Euler method as time integrator. We integrate in time the reduced order model with the BDF-q time stepping ($1\le q\le 5$) and prove optimal rate of convergence of order $q$ in time. Our set of snapshots is obtained from finite element approximations to the original model problem computed at different times. These finite element approximations can be obtained with any time integrator. The POD method is based on first order difference quotients of the snapshots. The reason for doing this is twofold. On the one hand, the use of difference quotients allow us to provide pointwise-in-time error bounds. On the other, the use of difference quotients is essential to get the expected rate $q$ in time since we apply that the BDF-q time stepping, $1\le q\le 5$, can be written as a linear combination of first order difference quotients.

Using BDF schemes in the temporal integration of POD-ROM methods

Abstract

In this paper we consider the numerical approximation of a semilinear reaction-diffusion model problem (PDEs) by means of reduced order methods (ROMs) based on proper orthogonal decomposition (POD). We focus on the time integration of the fully discrete reduced order model. Most of the analysis in the literature has been carried out for the implicit Euler method as time integrator. We integrate in time the reduced order model with the BDF-q time stepping () and prove optimal rate of convergence of order in time. Our set of snapshots is obtained from finite element approximations to the original model problem computed at different times. These finite element approximations can be obtained with any time integrator. The POD method is based on first order difference quotients of the snapshots. The reason for doing this is twofold. On the one hand, the use of difference quotients allow us to provide pointwise-in-time error bounds. On the other, the use of difference quotients is essential to get the expected rate in time since we apply that the BDF-q time stepping, , can be written as a linear combination of first order difference quotients.

Paper Structure

This paper contains 7 sections, 8 theorems, 75 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. BDF methods
  4. Proper orthogonal decomposition
  5. Error analysis of the POD method
  6. Numerical Experiments
  7. Conclusions

Key Result

lemma 1

Let $\delta(\zeta)=\sum_{i=0}^q \delta_i \zeta^i$ and $\mu(\zeta)=\sum_{j=0}^q\mu_j\zeta^j$ be polynomials that have no common divisor. Assume the following condition holds Let $(\cdot,\cdot)$ be a semi-inner product on a Hilbert space $H$ with associated norm $|\cdot|$. Then, there exists a symmetric positive-definite matrix $G=[g_{i,j}]\in {\Bbb R}^{q\times q}$ such that for $v_0,\ldots,v_q\in

Figures (2)

  • Figure 1: Components $u$ (up) and $v$ (down) of the periodic solution at time $t=0, t=2.3578 \approx T/3, t=4.7225 \approx 2T/3$ and $t=6.5159$, where the maximum of $\|\nabla u \|_0^2 + \|\nabla v\|_0^2$ is attained.
  • Figure 2: Maximum error along one period in the $L^2$ norm, $\max\limits_{q\leq n \leq M}\|u_r^s(t_n) - u_r^{bdf_q}(t_n)\|_0,$ (left) and $H_0^1$ norm, $\max\limits_{q\leq n \leq M}\|\nabla(u_r^s(t_n) - u_r^{bdf_q}(t_n))\|_0,$ (right) for $M=64, 128, 256, 512$ and $1024$.

Theorems & Definitions (15)

  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • theorem 4
  • ...and 5 more