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.

The relation between wetting properties and geometric parameters of fractal surfaces are widely discussed on the literature and, however, there are still divergences on this topic. Here we propose a simple theoretical model to describe the wetting properties of a droplet of water placed on a hierarchical structured surface and test the predictions of the model and the dependence of the droplet wetting state on the initial conditions using simulation of the 3-spin Potts model. We show that increasing the auto-similarity level of the hierarchy – called n – does not affect considerably the stable wetting state of the droplet but increases its contact angle. Simulations also explicit the existence of metastable states on this type of surfaces and shows that, when n increases, the metastability becomes more pronounced. Finally we show that the fractal dimension of the surface is not a good predictor of the contact angle of the droplet.

Abstract We design and implement an effective fully discrete Lagrangian–Eulerian scheme for a class of scalar, local and nonlocal models, and systems of hyperbolic problems in 1D. We propose statements, via a weak asymptotic analysis, which include existence, uniqueness, regularity, and numerical approximations of entropy-weak solutions computed with the scheme for the corresponding nonlinear initial value problem for the local scalar case. We study both convergence and weak bounded variation (BV) properties of the scheme to the entropy solution (for the local and scalar case) in the sense of Kruzhkov. The approach is based on the improved concept of no-flow curves, as introduced by the authors, and we highlight the strengths of the method: (i) the scheme for systems of hyperbolic problems does not require computation of the eigenvalues (exact or approximate) either to the numerical flux function or the weak CFL stability condition (wCFL) and (ii) we prove the properties: positivity-preserving, total variation nonincreasing, and maximum principle subject to the wCFL. We present numerical experiments to evaluate the shock capturing capabilities of the scheme in resolving the main features for hyperbolic problems: shock waves, rarefaction waves, contact discontinuities, positivity-preserving properties, and nonlinear wave formations and interactions.