1,720,973 research outputs found
Conservation Laws with Time Dependent Discontinuous Coefficients
No full textWe consider scalar conservation laws where the flux function depends discontinuously
on both the spatial and temporal locations. Our main results are the existence and well-posedness of
an entropy solution to the Cauchy problem. The existence is established by showing that a sequence
of front tracking approximations is compact in L1 , and that the limits are entropy solutions. Then,
using the definition of an entropy solution taken from [K. H. Karlsen, N. H. Risebro, and J. D.
Towers, Skr. K. Nor. Vidensk. Selsk., 3 (2003), pp. 1–49], we show that the solution operator is L1
contractive. These results generalize the corresponding results from [S. N. Kruˇzkov, Math. USSR-Sb.,
10 (1970), pp. 217–243] and also partially those from Karlsen, Risebro, and Towers
Convergence of an Engquist Osher scheme for a multidimensional triangular system of conservation laws
No full textWe consider a multi-dimensional triangular system of conservation
laws. This system arises as a model of three-phase flow in porous media and
includes multi-dimensional conservation laws with discontinuous coefficients
as a special case. The system is neither strictly hyperbolic nor symmetric.
We propose an Engquist-Osher type scheme for this system and show that the
approximate solutions generated by the scheme converge to a weak solution.
Numerical examples are also presented
A CONVERGENT DIFFERENCE SCHEME FOR A CLASS OF PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS MODELING PRICING UNDER UNCERTAINTY
No full textIn this paper we present a finite difference scheme to approximate viscosity solutions of a class of partial integro-differential equations describing pricing under model uncertainty. We establish that the approximations converge to the unique viscosity solution as the discretization parameter tends to zero, and give an asymptotic rate of the convergence. We also present several numerical examples showing this convergence
Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation
No full textRecent work \cite{Coclite:2005cr} has shown that the
Degasperis-Procesi equation is well-posed in the class of
(discontinuous) entropy solutions. In the present paper we
construct numerical schemes and prove that they converge to entropy
solutions. Additionally, we provide several numerical examples
accentuating that discontinuous (shock) solutions form independently
of the smoothness of the initial data. Our focus on discontinuous
solutions contrasts notably with the existing literature on the
Degasperis-Procesi equation, which seems to emphasize similarities
with the Camassa-Holm equation (bi-Hamiltonian structure,
integrabillity, peakon solutions, as the relevant functional space)
Analysis and numerical approximation of Brinkman regularization of two phase flows in porous media
No full textWe consider a hyperbolic-elliptic system of PDEs that arises in the modeling of two-phase flows in a porous medium. The phase velocities are modeled using a Brinkman regularization of the classical Darcy's law. We propose a notion of weak solutions for these equations and prove existence of these solutions. An efficient finite difference scheme is proposed and is shown to converge to the weak solutions of this system. The Darcy limit of the Brinkman regularization is studied numerically using the convergent finite difference scheme in two space dimensions as well as using both analytical and numerical tools in one space dimension. The results suggest that the Darcy limit of the Brinkmann regularization may not lead to the recovery of standard entropy solutions of the classical model for two-phase flows
An explicit finite difference scheme for the Camassa-Holm equation
No full textWe put forward and analyze an explicit finite difference
scheme for the Camassa-Holm shallow water equation that can handle
general H 1 initial data and thus peakon-antipeakon interactions. As-
suming a specified condition restricting the time step in terms of the
spatial discretization parameter, we prove that the difference scheme
converges strongly in H^1 towards a dissipative weak solution of the
Camassa-Holm equation
An explicit finite difference scheme for the Camassa-Holm equation
No full textWe put forward and analyze an explicit finite difference
scheme for the Camassa-Holm shallow water equation that can handle
general H 1 initial data and thus peakon-antipeakon interactions. As-
suming a specified condition restricting the time step in terms of the
spatial discretization parameter, we prove that the difference scheme
converges strongly in H^1 towards a dissipative weak solution of the
Camassa-Holm equation
A nonlocal Lagrangian traffic flow model and the zero-filter limit
Full text linkIn this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy
Some observations regarding the stationary Buckley–Leverett equation
No full textThe basic hyperbolic–elliptic black-oil model describes oil–water displacement in a porous
medium. Given its mathematical complexity, there is a need for particular simple solutions
for validation of numerical methods. We present a class of stationary solutions, which are easy
to compute, and in many cases are given by explicit formulae. These solutions are constructed
by a nonlinear coupling of two linear equations, an elliptic pressure equation and a hyperbolic
saturation equation
Well-posedness of the Initial Value Problem for the Ostrovsky–Hunter Equation with Spatially Dependent Flux
No full textIn this paper we study the Ostrovsky-Hunter equation for the case where the flux function f(x, u) may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (namely f(x)(x, u) is uniformly Lipschitz locally in u and f(u)(x, u) is uniformly bounded), then there exists a unique entropy solution. To show the existence, after proving some a priori estimates we have used the method of compensated compactness and to prove the uniqueness we have employed the method of doubling of variables
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