By Laurent Gosse
Substantial attempt has been drawn for years onto the advance of (possibly high-order) numerical innovations for the scalar homogeneous conservation legislations, an equation that is strongly dissipative in L1 due to surprise wave formation. this sort of dissipation estate is usually misplaced while contemplating hyperbolic structures of conservation legislation, or just inhomogeneous scalar stability legislation concerning accretive or space-dependent resource phrases, as a result of advanced wave interactions. An total weaker dissipation can show intrinsic numerical weaknesses via particular nonlinear mechanisms: Hugoniot curves being deformed via neighborhood averaging steps in Godunov-type schemes, low-order mistakes propagating alongside increasing features after having hit a discontinuity, exponential amplification of truncation blunders within the presence of accretive resource terms... This e-book goals at providing 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 really is divided into elements: one facing hyperbolic structures of stability legislation, corresponding to bobbing up from quasi-one dimensional nozzle stream computations, multiphase WKB approximation of linear Schrödinger equations, or gravitational Navier-Stokes structures. balance effects for viscosity suggestions of onedimensional stability legislation are sketched. the opposite being solely dedicated to the therapy of weakly nonlinear kinetic equations within the discrete ordinate approximation, reminiscent of those of radiative move, chemotaxis dynamics, semiconductor conduction, spray dynamics or linearized Boltzmann types. “Caseology” is among the major ideas utilized in those derivations. Lagrangian thoughts for filtration equations come to mind too. Two-dimensional tools 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
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This second order trick allows to solve the non-conservative jump relations associated to the non-conservative reformulation of the tumbling process at each interface, at the price of a stability criterion related to a stronger version of the non-resonance assumption though. Scattering matrices can be expressed in a rather straightforward manner, together with the correct way of implementing Neumann or Dirichlet boundary conditions at each edge of the bounded computational domain. As these scattering matrices appear as rank-one perturbations, specific inversion algorithms (the ShermanMorrison formula ) can be profitably implemented.