The aim of this work is to illustrate a recent contribution to the approximation of solutions to hyperbolic systems of balance laws, a class of partial differential equations that arise in many areas of application such as fluid mechanics. They describe the time evolution of certain bulk quantities, which are subject to a physical process involving both convection, and another mechanism (reaction, relaxation). In many situations, their large-time behavior is governed by an accurate balance between the transport terms and the other ones. It is then natural to design reliable numerical approximations in accordance with this qualitative property; by the property of being exact, or almost exact, when reproducing stationary solutions, they are often called "well-balanced". In this work we present an approach to the accuracy analysis of this class of schemes. The technique relies on the stability theory derived by Bressan, Liu, Yang and appears particularly effective in the assessment of the time dependence of the error. We focus on two basic situations: non-resonant scalar equations and semilinear systems with space-dependent damping. The work ends with some case studies, including a glance to a 2-dimensional situation.

Error Estimates for Well-Balanced Schemes on Simple Balance Laws. One-Dimensional Position-Dependent Models

AMADORI, DEBORA;
2015

Abstract

The aim of this work is to illustrate a recent contribution to the approximation of solutions to hyperbolic systems of balance laws, a class of partial differential equations that arise in many areas of application such as fluid mechanics. They describe the time evolution of certain bulk quantities, which are subject to a physical process involving both convection, and another mechanism (reaction, relaxation). In many situations, their large-time behavior is governed by an accurate balance between the transport terms and the other ones. It is then natural to design reliable numerical approximations in accordance with this qualitative property; by the property of being exact, or almost exact, when reproducing stationary solutions, they are often called "well-balanced". In this work we present an approach to the accuracy analysis of this class of schemes. The technique relies on the stability theory derived by Bressan, Liu, Yang and appears particularly effective in the assessment of the time dependence of the error. We focus on two basic situations: non-resonant scalar equations and semilinear systems with space-dependent damping. The work ends with some case studies, including a glance to a 2-dimensional situation.
978-3-319-24785-4
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/28570
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