Computing Qualitatively Correct Approximations of Balance by Laurent Gosse

By Laurent Gosse

Substantial attempt has been drawn for years onto the advance of (possibly high-order) numerical suggestions for the scalar homogeneous conservation legislation, an equation that's strongly dissipative in L1 because of surprise wave formation. this type of dissipation estate is usually misplaced whilst contemplating hyperbolic structures of conservation legislation, or just inhomogeneous scalar stability legislation related to accretive or space-dependent resource phrases, as a result of complicated wave interactions. An total weaker dissipation can display intrinsic numerical weaknesses via particular nonlinear mechanisms: Hugoniot curves being deformed through neighborhood averaging steps in Godunov-type schemes, low-order error propagating alongside increasing features after having hit a discontinuity, exponential amplification of truncation blunders within the presence of accretive resource terms... This publication goals at offering rigorous derivations of other, also known as well-balanced, numerical schemes which reach reconciling excessive accuracy with a higher robustness even within the aforementioned accretive contexts. it truly is divided into elements: one facing hyperbolic platforms of stability legislation, resembling coming up from quasi-one dimensional nozzle circulate computations, multiphase WKB approximation of linear Schrödinger equations, or gravitational Navier-Stokes structures. balance effects for viscosity recommendations of onedimensional stability legislation are sketched. the opposite being completely dedicated to the remedy of weakly nonlinear kinetic equations within the discrete ordinate approximation, equivalent to those of radiative move, chemotaxis dynamics, semiconductor conduction, spray dynamics or linearized Boltzmann types. “Caseology” is among the major options utilized in those derivations. Lagrangian concepts for filtration equations spring to mind too. Two-dimensional equipment are studied within the context of non-degenerate semiconductor models.

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Additional resources for Computing Qualitatively Correct Approximations of Balance Laws: Exponential-Fit, Well-Balanced and Asymptotic-Preserving

Example text

23) holds for w(uε , aε ). TV(a ), which implies that the sequence of 0-Riemann invariants wε ∈ BVloc (R+ ∗ × R) uniformly. TV(a ). 24) c 1 (R+ × R) as ε → 0. 1. 21) remains strictly hyperbolic. 24) implies that uε is bounded in L∞ too, even if no dissipativity assumption has been made on the source term g. 2 Localization Process of the Source Term on a Discrete Lattice 29 assumption implies that the bulk of hyperbolic waves cannot remain forever inside an area where the source term is big. 19) with u0 ∈ [0, 1], then w(u, a) = 1 − exp(ln(1 − u) + a) = 1 − exp(a) + u exp(a).

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