Paulo Amorim

Paulo Amorim, Philippe G. LeFloch, Bawer Okutmustur

We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy dissipation bound, which take into account the manifold geometry accurately and generalize techniques and estimates that were known in the (flat) Euclidian setting, only. The strong convergence of the scheme then is then a consequence of the well-posed theory recently developed by Ben-Artzi and LeFloch for conservation laws on manifolds.

Paulo Amorim, Christine Bernardi, Philippe G. LeFloch

We consider a system of nonlinear wave equations with constraints that arises from the Einstein equations of general relativity and describes the geometry of the so-called Gowdy symmetric spacetimes on T3. We introduce two numerical methods, which are based on pseudo-spectral approximation. The first approach relies on marching in the future time-like direction and toward the coordinate singularity t=0. The second approach is designed from asymptotic formulas that are available near this singularity; it evolves the solutions in the past timelike direction from "final" data given at t=0. This backward method relies a novel nonlinear transformation, which allows us to reduce the nonlinear source terms to simple quadratic products of the unknown variables. Numerical experiments are presented in various regimes, including cases where "spiky" structures are observed as the coordinate singularity is approached. The proposed backward strategy leads to a robust numerical method which allows us to accurately simulate the long-time behavior of a large class of Gowdy spacetimes.

Paulo Amorim, Joao-Paulo Dias, Mario Figueira et al.

We investigate the coupling between the nonlinear Schrödinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schrödinger--Burgers system, when the reference solution is an entropy--satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.

Paulo Amorim, Rinaldo M. Colombo, Andreia Teixeira

We address the study of a class of 1D nonlocal conservation laws from a numerical point of view. First, we present an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are lead to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.

Paulo Amorim, Maria Soledad Aronna, Debora de Oliveira Medeiros

We present an epidemiological model for vector-borne diseases that includes within-host viral load and antibody dynamics using structured transport equations. By incorporating the internal dynamics into the infected and recovered host compartments, the formulation introduces nonlinearities and nonlocalities. We establish analytical properties, including well-posedness and mass conservation, and characterize its characteristic curves. Furthermore, we derive a simplified Uniform Host Response (UHR) model featuring delay-type terms. For both the full and UHR frameworks, the basic reproduction number is determined and shown to serve as a threshold for the existence of an endemic equilibrium, and is related to the linear stability of the disease-free state. Finally, numerical experiments, parameterized specifically for Dengue fever, demonstrate how within-host mechanisms influence population-level epidemiological outcomes.

Paulo Amorim, Mário Figueira

We prove the convergence in a strong norm of a finite difference semi-discrete scheme approximating a coupled Schrödinger--KdV system on a bounded domain. This system models the interaction of short and long waves. Since the energy estimates available in the continuous case do not carry over to the discrete setting, we rely on a suitably truncated problem which we prove reduces to the original one. We present some numerical examples to illustrate our convergence result.

Paulo Amorim, Mário Figueira

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.

Paulo Amorim, Philippe G. LeFloch, Wladimir Neves

We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L1 error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.

Paulo Amorim, Mário Figueira

We consider the numerical approximation of a system of partial differential equations involving a nonlinear Schrödinger equation coupled with a hyperbolic conservation law. This system arises in models for the interaction of short and long waves. Using the compensated compactness method, we prove convergence of approximate solutions generated by semi-discrete finite volume type methods towards the unique entropy solution of the Cauchy problem. Some numerical examples are presented.

Paulo Amorim, Matania Ben-Artzi, Philippe G. LeFloch

This paper investigates some properties of entropy solutions of hyperbolic conservation laws on a Riemannian manifold. First, we generalize the Total Variation Diminishing (TVD) property to manifolds, by deriving conditions on the flux of the conservation law and a given vector field ensuring that the total variation of the solution along the integral curves of the vector field is non-increasing in time. Our results are next specialized to the important case of a flow on the 2-sphere, and examples of flux are discussed. Second, we establish the convergence of the finite volume methods based on numerical flux-functions satisfying monotonicity properties. Our proof requires detailed estimates on the entropy dissipation, and extends to general manifolds an earlier proof by Cockburn, Coquel, and LeFloch in the Euclidian case.