By Debora Amadori, Laurent Gosse
ISBN-10: 3319247840
ISBN-13: 9783319247847
ISBN-10: 3319247859
ISBN-13: 9783319247854
This monograph offers, in an enticing and self-contained shape, options in keeping with the L1 balance idea derived on the finish of the Nineteen Nineties via A. Bressan, T.-P. Liu and T. Yang that yield unique errors estimates for so-called well-balanced numerical schemes fixing 1D hyperbolic structures of stability legislation. Rigorous mistakes estimates are provided for either scalar stability legislation and a position-dependent leisure method, in inertial approximation. Such estimates make clear why these algorithms in accordance 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 on the subject of the therapy of 2nd stability legislation. the entire fabric supplied during this e-book is very appropriate for the knowledge of well-balanced schemes and may give a contribution to destiny improvements.
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Additional resources for Error Estimates for Well-Balanced Schemes on Simple Balance Laws: One-Dimensional Position-Dependent Models
Example text
3). In terms of Riemann coordinates (a, w), invariant domains correspond simply to rectangles. 5) for some constants a¯ 1 < a¯ 2 , w¯ 1 < w¯ 2 . 6) do not blow up in finite intervals (see [2, Eq. 7). 1 Properties of the Riemann Invariant w(u, a) The above choice, in presence of isolated zeros of g, is convenient since it provides a continuous function u → w(u, a) across these points. Indeed let us assume, for simplicity, that g(u) has a finite number of zeroes, say {u¯ j } j=1 ... N0 being an increasing sequence; no specific sign is required for g outside the set of zeroes.
23). 2], at least in terms of dependence on a − b. 23). 4 Stability Estimates, Non Accretive Case In this subsection we focus on the case of a non-accretive source term, that is N = sup k(x)g (u) ≤ 0 x,u ∀ x, u. 13), Λ(t; U1 , U2 ) = x2 x1 +Lt W1 (t, x)| p(x)| + |q(t, x)| d x. 23), with the difference that the multiplying term eκ2 ρ is replaced by 1. 1 now follows. 32) for a suitable constant C , and for all t: 0 ≤ t ≤ (x2 − x1 )/L. 32) do not depend on time. 32) is yet provided, focusing on the special case of source term being only space-dependent: ∂t u + ∂x f (u) = k(x).
54 4 Lyapunov Functional for Inertial Approximations Clearly, if G(fr− , f + ) = 0, then J∗ = f + − fr− for every δ > 0. 26) (subscripts in f ± were dropped). One easily finds a particular solution for δ = 0, F(J0 , 0; f ± ) = 0 ⇔ J0 = f + − f − . This solution corresponds to the case where there is no zero-wave because δ = ar − a vanishes. Let us verify that the following property holds: 0= ∂F (J0 , 0; f ± ) = ∂J B + ∂ρ B (2f − + J0 , J0 ) − ∂J B − ∂ρ B (2f + − J0 , J0 ), ∂J but since 2f + − J0 = f + + f − = 2f − + J0 , this expression reduces to ∂F 1 (J0 , 0; f ± ) = 2∂ρ B(f + + f − , f + − f − ) = = 0.