By Laurent Gosse
Substantial attempt has been drawn for years onto the improvement of (possibly high-order) numerical strategies for the scalar homogeneous conservation legislation, an equation that is strongly dissipative in L1 because of surprise wave formation. this type of dissipation estate is mostly misplaced while contemplating hyperbolic structures of conservation legislation, or just inhomogeneous scalar stability legislation regarding accretive or space-dependent resource phrases, as a result of advanced wave interactions. An total weaker dissipation can exhibit intrinsic numerical weaknesses via particular nonlinear mechanisms: Hugoniot curves being deformed via neighborhood averaging steps in Godunov-type schemes, low-order blunders propagating alongside increasing features after having hit a discontinuity, exponential amplification of truncation blunders within the presence of accretive resource terms... This booklet goals at proposing rigorous derivations of alternative, also known as well-balanced, numerical schemes which achieve reconciling excessive accuracy with a better robustness even within the aforementioned accretive contexts. it truly is divided into components: one facing hyperbolic platforms of stability legislation, resembling bobbing up from quasi-one dimensional nozzle move computations, multiphase WKB approximation of linear Schrödinger equations, or gravitational Navier-Stokes platforms. balance effects for viscosity ideas of onedimensional stability legislation are sketched. the opposite being completely dedicated to the remedy of weakly nonlinear kinetic equations within the discrete ordinate approximation, resembling those of radiative move, chemotaxis dynamics, semiconductor conduction, spray dynamics or linearized Boltzmann versions. “Caseology” is among the major thoughts utilized in those derivations. Lagrangian strategies for filtration equations spring to mind too. Two-dimensional tools are studied within the context of non-degenerate semiconductor models.
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Extra info for Computing Qualitatively Correct Approximations of Balance Laws: Exponential-Fit, Well-Balanced and Asymptotic-Preserving
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