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Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate

Claire Chainais-Hillairet (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation u t ( x , t ) + div F ( x , t , u ( x , t ) ) = 0 with the initial condition u 0 L ( N ) . Passing to the limit in these schemes, we prove the existence of an entropy solution u L i n f t y ( N × + ) . Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in L l o c p ( N × + ) , 1 ≤ p ≤ +∞. Furthermore, if u 0 L BV l o c ( N ) , we show that u BV l o c ( N × + ) , which leads to an “ h 1 4 ” error estimate between the approximate and the entropy solutions (where h defines the size of the...

Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics

Mária Lukáčová-Medviďová, Jitka Saibertová (2006)

Applications of Mathematics

In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics...

Formation of Singularities for Weakly Non-Linear N×N Hyperbolic Systems

Boiti, Chiara, Manfrin, Renato (2001)

Serdica Mathematical Journal

We present some results on the formation of singularities for C^1 - solutions of the quasi-linear N × N strictly hyperbolic system Ut + A(U )Ux = 0 in [0, +∞) × Rx . Under certain weak non-linearity conditions (weaker than genuine non-linearity), we prove that the first order derivative of the solution blows-up in finite time.

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Raimund Bürger, Ricardo Ruiz, Kai Schneider, Mauricio Sepúlveda (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order...

Fundamental solutions and singular shocks in scalar conservation laws.

Emmanuel Chasseigne (2003)

Revista Matemática Complutense

We study the existence and non-existence of fundamental solutions for the scalar conservation laws ut + f(u)x = 0, related to convexity assumptions on f. We also study the limits of those solutions as the initial mass goes to infinity. We especially prove the existence of so-called Friendly Giants and Infinite Shock Solutions according to the convexity of f, which generalize the explicit power case f(u) = um. We introduce an extended notion of solution and entropy criterion to allow infinite shocks...

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