Ece H. Korkmaz, Samet Y. Kadıoğlu, and Ersin Özuğurlu presented their research at the "International Conference on Mathematics and Computers with Applications (MCWA 2025)," which took place in Istanbul, Turkey, between 16-18 July 2025.

This presentation examines how different averaging schemes influence numerical solutions to reaction-diffusion equations in heat transfer and glioma growth modeling. Addressing the challenge of solution sensitivity to averaging methods, particularly on coarse meshes or during transient phases, it employs a finite volume discretization based on an α-damping flux scheme that reduces dependence on the choice of averaging method.

The study investigates the influence of harmonic, arithmetic, and weighted averaging schemes on the numerical solution of reaction-diffusion equations through two representative applications: heat transfer and glioma growth. For the heat transfer problem, discontinuous initial conditions and highly nonlinear diffusion coefficients are considered, which induce sharp thermal fronts. Glioma growth is modeled using the Fisher-Kolmogorov reaction-diffusion equation with spatially varying diffusivity to capture heterogeneous tissue structures. The governing equations are discretized using the finite volume method, with time integration based on the Crank-Nicolson method. The Jacobian-Free Newton-Krylov method is employed to solve the resulting nonlinear equations. The research demonstrates that with sufficiently fine spatial resolution, all averaging schemes yield nearly identical numerical solutions. However, the sensitivity of solutions to the choice of averaging scheme becomes significant on coarse meshes, where the conservative α-damping flux correction proves effective in reducing this dependence. These findings underscore the importance of matching the averaging technique to both the physical context and computational resolution, suggesting that weighted schemes, together with the AD flux correction, offer a flexible and effective alternative for practical computations where mesh refinement may be computationally prohibitive.