Related papers: A geometric approach to error estimates for conser…
We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The schemes are based on a method of lines…
Conservation laws are computed for various nonlinear partial differential equations that arise in elasticity and acoustics. Using a scaling homogeneity approach, conservation laws are established for two models describing shear wave…
Since the celebrated theorem of Lax and Wendroff, we know a necessary condition that any numerical scheme for hyperbolic problem should satisfy: it should be written in flux form. A variant can also be formulated for the entropy. Even…
It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar…
Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy…
A set of exact quasi-local conservation equations is obtained in the (1+1)-dimensional description of the Einstein's equations of (3+1)-dimensional spacetimes. These equations are interpreted as quasi-local energy, linear momentum, and…
The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose…
Stable numerical simulations for a hyperbolic system of conservation laws of relaxation type but not in divergence form are obtained by incorporating the physical entropy into the simulations. The entropy balance is utilized as an…
Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical…
We are concerned with a new solution formula and its applications to the analysis of properties of entropy solutions of the Cauchy problem for one-dimensional scalar hyperbolic conservation laws, wherein the flux functions exhibit convexity…
We present a dimensionally split method for solving hyperbolic conservation laws on Cartesian cut cell meshes. The approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux, and we provide a…
We show that in one space dimension Lipschitz solutions of extremal surface equations are equivalent to entropy solutions in $L^\infty(\R)$ of a non-strictly hyperbolic system of conservation laws. We obtain an explicit representation…
We introduce a generalization of Glimm's random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend…
In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. The two classes of SV methods are constructed by letting a piecewise $k$-th order ($k\ge…
We consider the initial value problem for a scalar conservation law in one space dimension with a single spatial flux discontinuity, the so-called two-flux problem. We prove that a well-known front tracking algorithm has a convergence rate…
We study the quasi-static limit for the $L^\infty$ entropy weak solution of scalar one-dimensional hyperbolic equations with strictly concave or convex flux and time dependent boundary conditions. The quasi-stationary profile evolves with…
Building on the information-theoretic perspective of P.~D.~Lax [\textit{Proc.\ Sympos., Math.\ Res.\ Center, Univ.\ Wisconsin}, 1978], we establish a two-sided quantitative compactness estimate for numerical solutions of scalar conservation…
We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely…
A posteriori estimates give bounds on the error between the unknown solution of a partial differential equation and its numerical approximation. We present here the methodology based on H1-conforming potential and H(div)-conforming…
This study proposes a novel spatial discretization procedure for the compressible Euler equations which guarantees entropy conservation at a discrete level when an arbitrary equation of state is assumed. The proposed method, based on a…