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A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport

Marius Paul Bruchhäuser, Markus Bause

TL;DR

The paper develops a cost-efficient, space-time adaptive algorithm for a coupled flow and convection-dominated transport problem using a multirate time discretization and goal-oriented error control based on the Dual Weighted Residual (DWR) method. By separating temporal and spatial errors and employing weighted indicators for the transport problem while using auxiliary indicators for the flow, the method achieves accurate, efficient adaptivity on space-time slabs. The approach is demonstrated on numerical examples showing robust performance, effective control of the final-time goal functional, and improved handling of convection-dominated fronts through SUPG stabilization. The work provides a practical framework for multi-physics simulations with strongly differing time scales and offers insights into extending DWR-based adaptivity to other coupled systems.

Abstract

In this work, a cost-efficient space-time adaptive algorithm based on the Dual Weighted Residual (DWR) method is developed and studied for a coupled model problem of flow and convection-dominated transport. Key ingredients are a multirate approach adapted to varying dynamics in time of the subproblems, weighted and non-weighted error indicators for the transport and flow problem, respectively, and the concept of space-time slabs based on tensor product spaces for the data structure. In numerical examples the performance of the underlying algorithm is studied for benchmark problems and applications of practical interest. Moreover, the interaction of stabilization and goal-oriented adaptivity is investigated for strongly convection-dominated transport.

A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport

TL;DR

The paper develops a cost-efficient, space-time adaptive algorithm for a coupled flow and convection-dominated transport problem using a multirate time discretization and goal-oriented error control based on the Dual Weighted Residual (DWR) method. By separating temporal and spatial errors and employing weighted indicators for the transport problem while using auxiliary indicators for the flow, the method achieves accurate, efficient adaptivity on space-time slabs. The approach is demonstrated on numerical examples showing robust performance, effective control of the final-time goal functional, and improved handling of convection-dominated fronts through SUPG stabilization. The work provides a practical framework for multi-physics simulations with strongly differing time scales and offers insights into extending DWR-based adaptivity to other coupled systems.

Abstract

In this work, a cost-efficient space-time adaptive algorithm based on the Dual Weighted Residual (DWR) method is developed and studied for a coupled model problem of flow and convection-dominated transport. Key ingredients are a multirate approach adapted to varying dynamics in time of the subproblems, weighted and non-weighted error indicators for the transport and flow problem, respectively, and the concept of space-time slabs based on tensor product spaces for the data structure. In numerical examples the performance of the underlying algorithm is studied for benchmark problems and applications of practical interest. Moreover, the interaction of stabilization and goal-oriented adaptivity is investigated for strongly convection-dominated transport.
Paper Structure (18 sections, 2 theorems, 58 equations, 10 figures, 1 table)

This paper contains 18 sections, 2 theorems, 58 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Variational Formulation and Discretization
  3. Variational Formulation
  4. Multirate and Discretization in Space and Time
  5. A Posteriori Error Estimation for Coupled Flow and Transport
  6. An A Posteriori Error Estimator for the Flow Problem
  7. Proof.
  8. An A Posteriori Error Estimator for the Transport Problem
  9. Proof.
  10. Algorithm and Practical Aspects
  11. Error Indicators and Approximation of Weights
  12. Implementation of Multirate Aspects
  13. Algorithm
  14. Numerical Examples
  15. Example 1 (Space-Time Adaptivity Studies for the Coupled Problem)
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $\{\mathbf{u},\mathbf{z}\} \in \mathcal{Y} \times \mathcal{Y}$, $\{\mathbf{u}_{\sigma},\mathbf{z}_{\sigma}\} \in \mathcal{Y}_{\sigma}^{r} \times \mathcal{Y}_{\sigma}^{r}$, and $\{\mathbf{u}_{\sigma h},\mathbf{z}_{\sigma h}\} \in \mathcal{Y}_{\sigma h}^{r,p_{\mathbf{v}},p_p} \times \mathcal{Y}_{\ Then, for the discretization errors in space and time we get the representation formulas Here, $\{

Figures (10)

  • Figure 4.2: Exemplary initialization of different temporal meshes for flow and transport.
  • Figure 4.3: Exemplary initialization of different spatial meshes for the flow ($\mathcal{T}_{h,n}^{\textrm{f}}$) and transport ($\mathcal{T}_{h,n}^{\textrm{t}}$) problem organized in a patch-wise manner.
  • Figure 4.4: Exemplary solution mesh transfer from a flow slab $Q_n^{\textrm{f}}$ (discontinuous Galerkin dG($0$) time discretization generated with one Gaussian quadrature point) to transport slabs $Q_n^{\textrm{t}}$ and $Q_{n+1}^{\textrm{t}}$ (discontinuous Galerkin dG($2$) time discretization generated with three Gaussian quadrature points), respectively. Here, each of the illustrated slabs consists of one cell in time and an independent and adaptively refined spatial triangulation.
  • Figure 5.1: Distribution of the temporal step size $\tau_K$ of the transport (adaptive, based on the DWR method) and $\sigma_K$ of the flow problem (global) over the time interval $I=(0,T]$, exemplary after 7 and 18 DWR-loops, corresponding to Table \ref{['table:1:L2final-KB2BR12-2232']}.
  • Figure 5.2: Comparison of adaptive spatial meshes at selected specific time points for the flow (based on the Kelly Error Estimator) (top, (a)--(d)) and transport (based on the DWR method) (bottom, (e)--(h)) problem, respectively, corresponding to the final loop in Table \ref{['table:1:L2final-KB2BR12-2232']}.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Remark 4.1