MECHANISMS FOR ERROR PROPAGATION AND CANCELLATION IN GLIMM'S SCHEME WITHOUT RAREFACTIONS

2007 ◽  
Vol 04 (03) ◽  
pp. 501-531 ◽  
Author(s):  
M. LAFOREST
Keyword(s):  
Error Propagation ◽  
Method Of Characteristics ◽  
Approximate Solutions ◽  
Posteriori Error ◽  
Scalar Conservation Laws ◽  
Wave Interactions ◽  
Single Wave ◽  
A Posteriori Error ◽  
Scalar Conservation ◽  
Norm Of The Error

We derive an a posteriori error bound for Glimm's approximate solutions to convex scalar conservation laws containing only shock waves. Using Liu's wave-tracing method, we show that the L1 norm of the error is bounded by a sum of residuals containing independent contributions from each wave in the approximate solution. We introduce a framework, similar to the method of characteristics, for the analysis of the local errors generated by wave interactions. The analysis allows for explicit cancellation among the errors created by a single wave and for error propagation along discontinuities.

A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

2005 ◽  
Vol 15 (07) ◽  
pp. 1119-1139 ◽  
Author(s):  
RODOLFO ARAYA ◽  
ABNER H. POZA ◽  
ERNST P. STEPHAN
Keyword(s):  
Error Estimate ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Diffusion Reaction ◽  
Posteriori Error Estimate ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Advection Diffusion ◽  
Norm Of The Error ◽  
Reaction Problem

In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition.

scholarly journals Non-classical Riemann solvers with nucleation

2004 ◽  
Vol 134 (5) ◽  
pp. 961-984 ◽  
Author(s):  
P. G. LeFloch ◽  
M. Shearer
Keyword(s):  
Propagation Speed ◽  
Riemann Solver ◽  
Scalar Conservation Laws ◽  
Wave Interactions ◽  
Kinetic Relation ◽  
Riemann Solvers ◽  
Flux Function ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
Film Flows

We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.

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