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- W4249811516 abstract "Consider an n × n hyperbolic system of conservation laws of the form ut + f(u)x = 0, (t,x) ∈ ℝ+ × ℝ, u ∈ ℝn. Here u = (u1,…,un) is the vector of conserved quantities, while the components of f = (f1,…,fn) are the luxes. The system is said strictly hyperbolic if at each point u the Jacobian matrix Df(u) has n real, distinct eigenvalues λ1(u)< ··· < λn(u). A fundamental ingredient to prove existence and stability in BV is the introduction of a functional, the Glimm–Liu interaction functional, which controls the interactions among non linear waves. Aim of this note is to present a simple interpretation of (the scalar part of) the interaction functional, and show how it can be extended to the following equation: a parabolic equation of the form ut + f(u)x = uxx; scalar semidiscrete schemes, for example the upwind scheme ut(t,x) + (f(u(t,x)) − f(u(t,x −1)) = 0, or the backward scheme (u(t,x) − u(t −1,x)) + f(u(t,x)) = 0; 2 × 2 relaxation approximation, in particular ut + vx = 0, vt + ux = f(u) − v. All these approximations are interesting from the physical and numerical point of view. Finding an interaction is one of the key steps toward the proof of BV bounds." @default.
- W4249811516 created "2022-05-12" @default.
- W4249811516 creator A5020113622 @default.
- W4249811516 date "2009-11-09" @default.
- W4249811516 modified "2023-09-24" @default.
- W4249811516 title "Singular Approximations to Hyperbolic Systems of Conservation Laws" @default.
- W4249811516 doi "https://doi.org/10.4171/009-1/4" @default.
- W4249811516 hasPublicationYear "2009" @default.
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