Analysis of a one-dimensional free boundary flow problem

A one-dimensional free surface problem is considered. It consists in Burgers’ equation with an additional diffusion term on a moving interval. The well-posedness of the problem is investigated and existence and uniqueness results are obtained locally in time. A semi-discretization in space with a piecewise linear finite element method is considered. A priori and a posteriori error estimates are given for the semi-discretization in space. A time splitting scheme allows to obtain numerical results in agreement with the theoretical investigations.Supported by the Swiss National Science Foundation     

Conservation law with the flux function discontinuous in the space variable—II: Convex–concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex–concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine–Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine–Hugoniot condition by a generalized Rankine–Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L¹$L^1$. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented.     

An Engquist-Osher type finite difference scheme with a discontinuous flux function in space

An Engquist-Osher type finite difference scheme is derived for dealing with scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient. The new monotone difference scheme is based on introducing a new interface numerical flux function, which is called a generalized Engquist-Osher flux. By means of this scheme, the existence and uniqueness of weak solutions to the scalar conservation laws are obtained and the convergence theorem is established. Some numerical examples are presented and the corresponding numerical results are displayed to illustrate the efficiency of the methods.     

Space–time discontinuous Galerkin finite element method for shallow water flows

A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.     

Exponential time integration for fast finite element solutions of some financial engineering problems

We consider exponential time integration schemes for fast numerical pricing of European, American, barrier and butterfly options when the stock price follows a dynamics described by a jump-diffusion process. The resulting pricing equation which is in the form of a partial integro-differential equation is approximated in space using finite elements. Our methods require the computation of a single matrix exponential and we demonstrate using a wide range of numerical tests that the combination of exponential integrators and finite element discretisations with quadratic basis functions leads to highly accurate algorithms for cases when the jump magnitude is Gaussian. Comparison with other time-stepping methods are also carried out to illustrate the effectiveness of our methods.     

Error estimate for finite volume scheme

We study the convergence of a finite volume scheme for the linear advection equation with a Lipschitz divergence-free speed in R ^d . We prove a h ^1/2-error estimate in the L ^∞(0,t;L ¹)-norm for BV data. This result was expected from numerical experiments and is optimal.     

A multiresolution finite volume scheme for two-dimensional hyperbolic conservation laws

In this paper, high-resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two-dimensional hyperbolic conservation laws. The method is conservative. Moreover, it is stable which is proven numerically in this paper. The computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed. Using this proposed scheme, we compute several two-dimensional model problems and a compressive rate ranging from about 5–10 is observed in all simulations.     

A fourth-order numerical method for the planetary geostrophic equations with inviscid geostrophic balance

The planetary geostrophic equations with inviscid balance equation are reformulated in an alternate form, and a fourth-order finite difference numerical method of solution is proposed and analyzed in this article. In the reformulation, there is only one prognostic equation for the temperature field and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition. The key observation is that all the velocity profiles can be explicitly determined by the temperature gradient, by utilizing the special form of the Coriolis parameter. This brings convenience and efficiency in the numerical study. In the fourth-order scheme, the temperature is dynamically updated at the regular numerical grid by long-stencil approximation, along with a one-sided extrapolation near the boundary. The velocity variables are recovered by special solvers on the 3-D staggered grid. Furthermore, it is shown that the numerical velocity field is divergence-free at the discrete level in a suitable sense. Fourth order convergence is proven under mild regularity requirements. R. Samelson was supported by NSF grant OCE04-24516 and Navy ONR grant N00014-05-1-0891. R. Temam was supported by NSF grant DMS-0604235 and the research fund of Indiana University. S. Wang was supported by NSF grant DMS-0605067 and Navy ONR grant N00014-05-1-0218.     

Staircasing in semidiscrete stabilised inverse linear diffusion algorithms

We consider a semidiscrete model problem for the approximation of stabilised inverse linear diffusion processes. The work is motivated by an important observation on fully discrete schemes concerning the so-called staircasing phenomenon: when sharpening monotone data profiles, fully discrete methods generally introduce stepfunction-type solutions reminiscent of staircases. In this work, we show by an analysis of dynamical systems in corresponding semidiscrete formulations that already the semidiscrete numerical model contains the relevant information on the occurrence of staircasing. Numerical experiments confirm and complement the theoretical results.     

An adaptive numerical method to handle blow-up in a parabolic system

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments that illustrate the behaviour of the adaptive method.     

Uniqueness of the bounded solution to a strongly degenerate parabolic problem

This paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem. The equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies. By using Holmgren’s approach, we prove that the bounded solution of the associated initial–boundary value problem is unique under some essentially necessary condition on the growth of the convection.     

An investigation of the internal structure of shock profiles for shock capturing schemes

The theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete. Previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions, so called continuum shock profiles. However, the previous papers have focused on linear methods. We have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes, including Riemann solver based flux limiter methods, high resolution central schemes and ENO type methods. In some cases, no continuum profiles exists. However, in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle. The possibility of such behavior was anticipated in Bultelle, Grassin and Serre, 1998, but no specific examples, or other evidence, of this behavior have previously been given. In other cases, our results indicate that continuum shock profiles exist, but are very complicated. We also study phase space orbits with regard to post shock oscillations.     

Numerical approximation of the Cahn-Larché equation

Summary Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation - a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account. Here, we present a discretization based on finite elements and an implicit Euler scheme. We first show solvability and uniqueness of solutions. Based on an energy decay property we then prove convergence of the scheme. Finally we present numerical experiments showing the impact of elasticity on the morphology of the microstructure.Research supported by DFG Priority Program Analysis, Modeling and Simulation of Multiscale Problems under AR234/5-2 and GA695/2-2     

Godunov-type approximation for a general resonant balance law with large data

We consider the Cauchy problem for the 2×2 nonstrictly hyperbolic system

     

Entropy solutions for a fourth-order nonlinear degenerate problem for noise removal

In this paper, we establish the existence and uniqueness of entropy solutions for a fourth-order nonlinear degenerate parabolic problem for noise removal in dimension 1≤d<4$1\le d<4$.     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework

We present a relaxation system for ideal magnetohydrodynamics (MHD) that is an extension of the Suliciu relaxation system for the Euler equations of gas dynamics. From it one can derive approximate Riemann solvers with three or seven waves, that generalize the HLLC solver for gas dynamics. Under some subcharacteristic conditions, the solvers satisfy discrete entropy inequalities, and preserve positivity of density and internal energy. The subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate Riemann solver itself. The 7-wave version of the solver is able to resolve exactly all material and Alfven isolated contact discontinuities. Practical considerations and numerical results will be provided in another paper.     

High order compact Alternating Direction Implicit method for the generalized sine-Gordon equation

High order compact Alternating Direction Implicit scheme is given for solving the generalized sine-Gordon equation in a two-dimensional rectangular domain. We apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives, and we give a linearized three time level algorithm for solving the original nonlinear equation. Error estimate is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this algorithm.     

Global conservative solutions for a model equation for shallow water waves of moderate amplitude

In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in [11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin [2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data.