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A finite-difference summation-by-parts, conditionally stable partitioned algorithm for conjugate heat transfer problems

Sarah Nataj, David C. Del Rey Fernández, David Brown, Rajeev Jaiman

TL;DR

This work develops a high-order finite-difference partitioned solver for conjugate heat transfer problems by integrating summation-by-parts (SBP) operators with simultaneous approximation terms (SATs) on curvilinear grids. The left and right subdomains solve fluid-like and solid-like equations, respectively, and exchange interface data through carefully designed SAT penalties to yield provable conditional energy stability under a CFL-like time-step constraint and parameter choices. The authors extend the 1D SBP–SAT stability analysis to 3D curvilinear geometries, introduce time-extrapolation-based interface initial guesses (EXT) to improve accuracy, and demonstrate numerical verification with manufactured solutions, highlighting spatial convergence and interface behavior. The approach enables truly modular solvers for multi-physics problems on complex geometries while maintaining energy stability and high-order accuracy, with potential extensions to non-conforming meshes and more realistic turbulent transfer scenarios.

Abstract

In this work, we design and analyze a novel, provably conditionally stable, weakly coupled partitioned scheme to solve the conjugate heat transfer (CHT) problem. We consider a model CHT problem consisting of linear advection-diffusion and heat equations, coupled at an interface through continuity of temperature and heat flux. We employ high-order summation-by-parts finite-difference operators in conjunction with simultaneous-approximation-terms (SATs) in curvilinear coordinates for spatial derivatives, combined with first- and second-order time discretizations and temporal extrapolation at the interface. Energy stability is maintained by carefully selecting SAT parameters at the interface. A range of coupling parameters are explored to identify those that yield a stable scheme, and a stepwise approach for choosing SAT parameters that ensure stability is given. The effectiveness of the method is demonstrated through numerical experiments in a two-dimensional model problem on a rectangular domain with curvilinear grids. The proposed approach enables the development of high-order, conditionally stable partitioned solvers suitable for general geometries.

A finite-difference summation-by-parts, conditionally stable partitioned algorithm for conjugate heat transfer problems

TL;DR

This work develops a high-order finite-difference partitioned solver for conjugate heat transfer problems by integrating summation-by-parts (SBP) operators with simultaneous approximation terms (SATs) on curvilinear grids. The left and right subdomains solve fluid-like and solid-like equations, respectively, and exchange interface data through carefully designed SAT penalties to yield provable conditional energy stability under a CFL-like time-step constraint and parameter choices. The authors extend the 1D SBP–SAT stability analysis to 3D curvilinear geometries, introduce time-extrapolation-based interface initial guesses (EXT) to improve accuracy, and demonstrate numerical verification with manufactured solutions, highlighting spatial convergence and interface behavior. The approach enables truly modular solvers for multi-physics problems on complex geometries while maintaining energy stability and high-order accuracy, with potential extensions to non-conforming meshes and more realistic turbulent transfer scenarios.

Abstract

In this work, we design and analyze a novel, provably conditionally stable, weakly coupled partitioned scheme to solve the conjugate heat transfer (CHT) problem. We consider a model CHT problem consisting of linear advection-diffusion and heat equations, coupled at an interface through continuity of temperature and heat flux. We employ high-order summation-by-parts finite-difference operators in conjunction with simultaneous-approximation-terms (SATs) in curvilinear coordinates for spatial derivatives, combined with first- and second-order time discretizations and temporal extrapolation at the interface. Energy stability is maintained by carefully selecting SAT parameters at the interface. A range of coupling parameters are explored to identify those that yield a stable scheme, and a stepwise approach for choosing SAT parameters that ensure stability is given. The effectiveness of the method is demonstrated through numerical experiments in a two-dimensional model problem on a rectangular domain with curvilinear grids. The proposed approach enables the development of high-order, conditionally stable partitioned solvers suitable for general geometries.
Paper Structure (18 sections, 9 theorems, 164 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 18 sections, 9 theorems, 164 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The continuous system and well-posedness
  3. 1D Cartesian SBP–SAT formulation, monolithic and partitioned interface coupling
  4. 1D discretization and interface coupling using SBP-SATs
  5. Energy stability of the discrete 1D monolithic and partitioned scheme
  6. Partitioned SBP-SAT scheme in 3D curvilinear setting
  7. Coordinate transformation
  8. Transforming the PDE to curvilinear coordinates
  9. Multi-dimensional SBP operators, governing equations and boundary and interface SATs in 3D curvilinear coordinates
  10. SBP--SAT partitioned scheme in 3D curvilinear coordinates
  11. Discrete stability analysis of partitioned scheme in the 3D curvilinear setting
  12. Using extrapolation in time of order two at interface as initial guess
  13. Applying second-order time discretization
  14. Numerical verification
  15. Conclusion
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\,\overline{\mathsf{P}}_{}=\mathop{\mathrm{diag}}\nolimits(\hat{p}_1,\ldots,\hat{p}_{N_x})$ with $\hat{p}_i>0$. Define Then for any grid vector $\hbox{\boldmath $z$}\in\mathbb{R}^{N_x}$, where $\|\hbox{\boldmath $z$}\|^2_{\overline{\mathsf{P}}_{}}:=\hbox{\boldmath $z$}^{\mathrm{T}}\overline{\mathsf{P}}_{}\hbox{\boldmath $z$}$.

Figures (6)

  • Figure 1: Domain representation for the model CHT problem in 2D.
  • Figure 2: Sequencing of coupling scheme for BE-EXT2 at the beginning of the loop.
  • Figure 3: The distribution of nodes on physical (left) and reference (right) domains
  • Figure 4: Results of spatial convergence study of iterated partitioned scheme ($N_{\ell oop}=2$) in comparison with monolithic scheme with time discretization BEFE where extrapolation in time of order 2 (EXT2) is used at interface. The results are given at final time $T=1$ with time step $\delta t= 10^{-4}$.
  • Figure 5: Comparison of the convergence of monolithic scheme, iterated BEFE-EXT1 and non-iterated BEFE-EXT2, $\delta t=10^{-4}$ and $p=3$.
  • ...and 1 more figures

Theorems & Definitions (20)

  • definition thmcounterdefinition
  • Lemma 1: Discrete trace inequality in 1D
  • proof
  • Theorem 2: Stability of the monolithic 1D scheme
  • proof
  • Theorem 3: Stability of the partitioned 1D scheme
  • proof
  • Definition 4
  • Lemma 5
  • proof
  • ...and 10 more