By Laurent Gosse
Substantial attempt has been drawn for years onto the advance of (possibly high-order) numerical recommendations for the scalar homogeneous conservation legislations, an equation that is strongly dissipative in L1 because of surprise wave formation. this kind of dissipation estate is usually misplaced whilst contemplating hyperbolic platforms 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 via neighborhood averaging steps in Godunov-type schemes, low-order error propagating alongside increasing features after having hit a discontinuity, exponential amplification of truncation mistakes within the presence of accretive resource terms... This booklet goals at featuring rigorous derivations of alternative, also known as well-balanced, numerical schemes which achieve reconciling excessive accuracy with a higher robustness even within the aforementioned accretive contexts. it's divided into components: one facing hyperbolic platforms of stability legislation, resembling coming up from quasi-one dimensional nozzle movement computations, multiphase WKB approximation of linear Schrödinger equations, or gravitational Navier-Stokes platforms. balance effects for viscosity strategies of onedimensional stability legislation are sketched. the opposite being fullyyt dedicated to the remedy of weakly nonlinear kinetic equations within the discrete ordinate approximation, corresponding to those of radiative move, chemotaxis dynamics, semiconductor conduction, spray dynamics or linearized Boltzmann types. “Caseology” is without doubt one of the major options 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
As soon as the field E or the parameter τ1 became big, very curved streamlines of flow are perturbed by the higher viscosity of the 2D scheme which cannot stabilize correctly. The issue is a consequence of the fact that even an optimum N-scheme approximates characteristic curves locally as straight lines: it is thus unable to let a particle make a U-turn in one time-step (the Hamiltonian-Preserving schemes of Chapter 6 can, actually). This means that the 2D approximation in the vicinity of the channel/drain junction cannot be as clean as it is with a 1D scheme (built for instance on ) which is able to represent more accurately shear flows in the x, v plane.
Proof. Let N ∈ C2 (R) be such that: N ≥ 0 and N(x) = |x| for |x| ≥ 1. We then define 0 a sequence N ε (x) = ε N( x−l ε ) converging in C (R) towards x → |x − l| as ε → 0. 6) holds for each N , ε > 0. Thanks to the L∞ bound, we get conver1 by means of Lebesgue’s dominated convergence theorem. 7), one picks |l| ≥ u L∞ . 1), that is to say, the resolution of an initial data consisting in a single jump uL , uR between constant states at the origin x = 0. The main difference with the homogeneous case k ≡ 0 is that its entropy solution ceases to be self-similar: one cannot express it in the form R(x/t; uL , uR ) any more.
On the relationship between cell balance equations for chemotactic cell populations. SIAM J. Applied Math. 52, 1426–1441 (1992) 32. : The Capriciousness of Numerical Methods for Singular Perturbations. SIAM Review 53, 157–173 (2011) 33. : Conditions for runaway phenomena in the kinetic theory of swarms. J. Math. Phys. 30, 1177–1186 (1989) 34. Gartland Jr. : On the uniform convergence of the Scharfetter-Gummel discretization in one dimension. SIAM J. Numer. Anal. 30, 749–758 (1993) 35. : A generalized Riemann problem for quasi one dimensional gas flows.