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

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.

💡 Research Summary

**
This paper introduces a novel, provably conditionally stable, weakly‑coupled partitioned algorithm for solving conjugate heat transfer (CHT) problems. The model CHT system consists of a linear advection‑diffusion equation in a fluid sub‑domain and a pure heat‑conduction equation in a solid sub‑domain, coupled through continuity of temperature and heat flux at a common interface. The authors employ high‑order summation‑by‑parts (SBP) finite‑difference operators together with simultaneous‑approximation‑terms (SAT) to discretize spatial derivatives on curvilinear grids, and they combine first‑ and second‑order time discretizations with temporal extrapolation at the interface.

The core contribution lies in the design of SAT penalty terms at the interface that guarantee energy stability for the partitioned scheme. By carefully selecting the penalty parameters, the authors derive discrete energy estimates that show a non‑increasing discrete energy under a CFL‑like condition linking the time step, spatial mesh size, and advection speed. The analysis is first carried out in a one‑dimensional Cartesian setting, where the SBP operators satisfy the discrete integration‑by‑parts property and the SAT terms enforce the interface conditions weakly. Two time integration strategies are examined: a first‑order backward Euler method and a second‑order scheme (Crank‑Nicolson or second‑order backward difference). In both cases, the scheme remains conditionally stable provided the penalty parameters lie within identified bounds and the time step satisfies the derived CFL restriction. Temporal extrapolation of interface values (first‑ or second‑order) is shown to preserve the stability property while improving temporal accuracy.

The methodology is then extended to three‑dimensional curvilinear coordinates. The authors construct SBP operators that respect discrete metric identities, ensuring that the geometric mapping does not destroy the SBP property. The same SAT framework is applied on curved interfaces, and analogous discrete energy estimates are derived, confirming that the conditional stability carries over to complex geometries.

Numerical experiments are performed on a two‑dimensional rectangular domain discretized with non‑uniform curvilinear grids. The fluid region is solved with the advection‑diffusion equation, the solid region with the heat equation, and the interface conditions are enforced via the proposed SAT terms. A systematic sweep of penalty parameters and time steps demonstrates that the theoretical stability limits are sharp: when the parameters satisfy the prescribed conditions, the discrete energy decays as expected; violating them leads to growth and instability. The partitioned algorithm achieves high‑order spatial accuracy comparable to a monolithic SBP‑SAT implementation, though a modest increase in error is observed near the interface unless a few sub‑iterations are performed. Using second‑order temporal extrapolation further improves the overall accuracy without compromising stability.

In summary, the paper delivers a high‑order, conditionally stable partitioned framework for CHT that can be applied on general curvilinear meshes and that allows existing single‑physics solvers to be reused. The step‑by‑step guideline for selecting SAT penalty parameters, together with the rigorous energy analysis, provides practitioners with a practical tool for robust multi‑physics simulations involving fluid–solid heat exchange. Future work may extend the approach to nonlinear material properties, moving interfaces, and hybrid strong/weak coupling strategies to further enhance accuracy and efficiency.