A note on the uniqueness of entropy solutions to first order quasilinear equations.
A positivity preserving central scheme for shallow water flows in channels with wet-dry states
We present a high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from...
A priori estimates for solutions of spinodal decomposition problem.
A result of existence for an original convection-diffusion equation.
En este artículo se estudia el análisis matemático de una ley de conservación que no es clásica. El modelo describe procesos estatigráficos en Geología y tiene en cuenta una condición de tasa de erosión limitada. En primer lugar se presentan el modelo físico y la formulación matemática (posiblemente nueva). Tras enunciar la definición solución se presentan las herramientas que permiten probar la existencia de soluciones.
A Riemann problem with small viscosity and dispersion.
A steady-state capturing method for hyperbolic systems with geometrical source terms
We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar...
A steady-state capturing method for hyperbolic systems with geometrical source terms
We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar...
A Systematic Approach for Correcting Nonlinear Instabilities. The Lax-Wendroff Scheme for Scalar Conservation Laws.
A uniqueness result for the continuity equation in two dimensions
We characterize the autonomous, divergence-free vector fields on the plane such that the Cauchy problem for the continuity equation admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential associated to . As a corollary we obtain uniqueness under the assumption that the curl of is a measure. This result can be extended to certain non-autonomous vector fields with bounded divergence....
Accurate numerical discretizations of non-conservative hyperbolic systems
We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating...
Accurate numerical discretizations of non-conservative hyperbolic systems
We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating...
Adaptive finite element relaxation schemes for hyperbolic conservation laws
We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...
Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws
We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...
An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws
We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov’s method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov’s method. It turns...
An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws
We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov's method. It turns...
An asymptotic expansion for the solution of the generalized Riemann problem. Part I : general theory
An asymptotic expansion for the solution of the generalized Riemann problem. Part 2 : application to the equations of gas dynamics
An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions
An existence result for balance laws with multifunctions: a model from the theory of granular media
We study an example of the balance law with a multifunction source term, coming from the theory of granular media. We prove the existence of "weak entropy solutions" to this system, using the vanishing viscosity method and compensated compactness. Because of the occurrence of a multifunction we give a new definition of the weak entropy solutions.
An Extension of the Lax-Richtmyer Theory.