Sarah Nataj | Searcharxiv

arXiv2026

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.

math.NA↗

arXiv2026

Space-time spectral element summation-by-parts method for heterogeneous transient heat conduction arising in topology optimization

We develop a stable space-time spectral element method for transient heat conduction in heterogeneous multi-material domains, motivated by topology optimization. The method treats space and time simultaneously, using summation-by-parts (SBP) operators in both directions and simultaneous approximation terms (SATs) to impose boundary, initial, terminal, and material-interface conditions weakly, yielding a stable monolithic space-time scheme on heterogeneous domains. Stability is proven under specific conditions on the SAT parameters, scaled with the spatial mesh resolution and material properties. We compute design sensitivities using a discrete space-time adjoint scheme that is dual-consistent with the primal SBP-SAT scheme. Numerical experiments demonstrate spectral convergence of the forward and adjoint solution errors and of the error in the functional output for smooth manufactured solutions in heterogeneous domains. Also under mesh refinement, the observed errors indicate higher-order convergence of the optimal objective value compared with the optimized design. We validate the resulting optimal design by comparison with an independently computed reference optimal design and report time-to-solution and cost-of-accuracy curves, comparing against low-order time-marching and all-at-once solvers for the forward and adjoint systems. The proposed scheme attains high accuracy with fewer space-time degrees of freedom and remains stable, reducing time-to-solution and memory compared with an alternative all-at-once solver. This makes it a future candidate for large-scale topology optimization of time-dependent thermal systems.

math.NA↗