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Spectral Variational Multi-Scale method for parabolic problems. Application to 1D transient advection-diffusion equations

Tomás Chacón Rebollo, Soledad Fernández-García, David Moreno-Lopez, Isabel Sánchez Muñoz

TL;DR

This work extends spectral Variational Multi-Scale (VMS) methods to parabolic problems by modeling sub-grid scales with eigenpairs of the local elliptic operator. The approach specializes to 1D transient advection-diffusion, where sub-grid components are computed exactly via a spectral expansion on each element, and a two-fold offline/online strategy reduces online cost. The authors prove that, for constant advection, the fully spectral VMS solution matches the implicit Euler semi-discretisation at Lagrange interpolation nodes, and demonstrate substantial accuracy improvements over stabilized methods in 1D tests, including large Peclet numbers. A feasible offline/online variant is developed to further enhance efficiency, with numerical results confirming stability, maximum-principle satisfaction, and improved accuracy, while indicating potential for extension to higher dimensions.

Abstract

In this work, we introduce a Variational Multi-Scale (VMS) method for the numerical approximation of parabolic problems, where sub-grid scales are approximated from the eigenpairs of associated elliptic operator. The abstract method is particularized to the one-dimensional advection-diffusion equations, for which the sub-grid components are exactly calculated in terms of a spectral expansion when the advection velocity is approximated by piecewise constant velocities on the grid elements. We prove error estimates that in particular imply that when Lagrange finite element discretisations in space are used, the spectral VMS method coincides with the exact solution of the implicit Euler semi-discretisation of the advection-diffusion problem at the Lagrange interpolation nodes. We also build a feasible method to solve the evolutive advection-diffusion problems by means of an offline/online strategy with reduced computational complexity. We perform some numerical tests in good agreement with the theoretical expectations, that show an improved accuracy with respect to several stabilised methods.

Spectral Variational Multi-Scale method for parabolic problems. Application to 1D transient advection-diffusion equations

TL;DR

This work extends spectral Variational Multi-Scale (VMS) methods to parabolic problems by modeling sub-grid scales with eigenpairs of the local elliptic operator. The approach specializes to 1D transient advection-diffusion, where sub-grid components are computed exactly via a spectral expansion on each element, and a two-fold offline/online strategy reduces online cost. The authors prove that, for constant advection, the fully spectral VMS solution matches the implicit Euler semi-discretisation at Lagrange interpolation nodes, and demonstrate substantial accuracy improvements over stabilized methods in 1D tests, including large Peclet numbers. A feasible offline/online variant is developed to further enhance efficiency, with numerical results confirming stability, maximum-principle satisfaction, and improved accuracy, while indicating potential for extension to higher dimensions.

Abstract

In this work, we introduce a Variational Multi-Scale (VMS) method for the numerical approximation of parabolic problems, where sub-grid scales are approximated from the eigenpairs of associated elliptic operator. The abstract method is particularized to the one-dimensional advection-diffusion equations, for which the sub-grid components are exactly calculated in terms of a spectral expansion when the advection velocity is approximated by piecewise constant velocities on the grid elements. We prove error estimates that in particular imply that when Lagrange finite element discretisations in space are used, the spectral VMS method coincides with the exact solution of the implicit Euler semi-discretisation of the advection-diffusion problem at the Lagrange interpolation nodes. We also build a feasible method to solve the evolutive advection-diffusion problems by means of an offline/online strategy with reduced computational complexity. We perform some numerical tests in good agreement with the theoretical expectations, that show an improved accuracy with respect to several stabilised methods.
Paper Structure (11 sections, 4 theorems, 74 equations, 15 figures, 3 tables)

This paper contains 11 sections, 4 theorems, 74 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Spectral VMS method
  3. Application to transient advection-diffusion problems
  4. One dimension problems
  5. Feasible method: offline/online strategy
  6. Application to 1D transient advection-diffusion problems
  7. Numerical Tests
  8. Test 1: Accuracy of spectral VMS method for constant advection velocity
  9. Test 2: Accuracy of spectral VMS method
  10. Test 3: Accuracy of the feasible spectral VMS method. Comparison with other stabilised methods
  11. Conclusions

Key Result

Theorem 2.1

Let us assume that there exists a complete sub-set $\{\tilde{z}_j^{n,K}\}_{j\in\mathbb{N}}$ on $\tilde{X}_K$ formed by eigenfunctions of the operator $\mathcal{L}^{n}_K$, which is an orthonormal system in $L^2_{p^{n,K}}(K)$ for some weight function $p^{n,K}\in C^1(\bar{K}).$ Then, where $\beta_j^{n,K} = (\Lambda_j^{n,K})^{-1}$, with $\Lambda_j^{n,K}=1+\Delta t \, \lambda_j^{n,K}$ being $\lambda_j

Figures (15)

  • Figure 1: Values of the spectral series to compute the diagonal coefficient of matrix $A_3$ for each pair $(P,S)$.
  • Figure 2: Values of the spectral series to compute the diagonal coefficient of matrix $A_3$ for $(P,S) \in (0,20)\times (0,1)$.
  • Figure 3: Values of the spectral series to compute the diagonal coefficient of matrix $A_3$ for $(P,S) \in (0,1)\times (0,1)$.
  • Figure 4: Number of summands needed to reach a first term with absolute value lower than $\varepsilon=10^{-10}$ for the series defining the diagonal coefficient of matrix $A_3$, in terms of $(P,S)$.
  • Figure 5: Splitting of interpolation cell for online computation of matrices coefficients.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Theorem 2.1
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof