A relaxation approach to modeling properties of hyperbolic–parabolic type models

In this work, we propose a novel relaxation modeling approach for partial differential equations (PDEs) involving convective and diffusive terms. We reformulate the original convection–diffusion problem as a system of hyperbolic equations coupled with relaxation terms. In contrast to existing literature on relaxation modeling, where the solution of the reformulated problem converges to certain types of equations in the diffusive limit, our formalism treats the augmented problem as a system of coupled hyperbolic equations with relaxation acting on both the convective flux and the source term. Furthermore, we demonstrate that the new system of equations satisfies Liu’s sub-characteristic condition. To verify the robustness of our proposed approach, we perform numerical experiments on various important models, including nonlinear convection–diffusion problems with discontinuous coefficients. The results show the promising potential of our relaxation modeling approach for both pure and applied mathematical sciences, with applications in different models and areas. •We propose a novel relaxation approach for modeling convection-diffusion problems via a hyperbolic system.•The new system satisfies Liu’s sub-characteristic condition, ensuring solution stability.•We apply the relaxation modeling to both classical and non-classical models.•From numerical experiments, we verify the new system’s solutions converge to those of the original equations.