By Debora Amadori, Laurent Gosse
This monograph provides, in an enticing and self-contained shape, options according to the L1 balance concept derived on the finish of the Nineties by means of A. Bressan, T.-P. Liu and T. Yang that yield unique blunders estimates for so-called well-balanced numerical schemes fixing 1D hyperbolic platforms of stability legislation. Rigorous mistakes estimates are awarded for either scalar stability legislation and a position-dependent rest process, in inertial approximation. Such estimates make clear why these algorithms in keeping with resource phrases dealt with like "local scatterers" can outperform different, extra normal, numerical schemes. Two-dimensional Riemann difficulties for the linear wave equation also are solved, with dialogue of the problems raised in terms of the therapy of second stability legislation. the entire fabric supplied during this booklet is very proper for the knowledge of well-balanced schemes and may give a contribution to destiny improvements.
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Additional info for Error Estimates for Well-Balanced Schemes on Simple Balance Laws: One-Dimensional Position-Dependent Models
18) The identity here above just asserts that (b(x), v(t, x)) and (a(x), ν(t, ˜ x)) belong to the same stationary curve. 4] provide an estimate on the L 1 norm of U1 − U2 in the domain of determinacy. Given x1 < x2 , x2 − x1 (t, x) : 0 ≤ t ≤ , x1 + Lt < x < x2 . 3), and P1 , P2 be their corresponding partitions with δ1 , δ2 > 0 the mesh parameters, respectively. 14) Λ(t) := Λ(t; U1 , U2 ) satisfies for all 0 ≤ s ≤ t ≤ (x2 − x1 )/L: Λ(t) − Λ(s) ≤ C · eκ2 ρ · [δ1r1 + δ2 r2 ]. 1, denote by I (t) the integral I (t) = x2 x1 +Lt |u(t, x) − v(t, x)| d x, 0≤t ≤ x2 − x1 .
5) A typical choice for g is given by the relaxation term: g(ρ, J) = A(ρ) − J. 4) perfectly matches the two-scale relaxation framework studied in . 6) would be for instance, g(x, ρ, J) = A(x, ρ) − J, x → A(x, ·) ∈ C 1 (R), sup |∂ρ A(x, ρ)| < 1. 1 A Class of Position-Dependent Semilinear Systems 47 with E(x) a small static electric field, and τ (x) standing for a space-dependent relaxation time depending on the local doping concentration. 8) where G(f − , f + ) := g f + + f − , f + − f − . 8) are such that f ± (t = 0, ·) = f0± ∈ L 1 ∩ BV (R).
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