Convergence of the Lambert-McLeod trajectory solver and of the celf method

Summary A trajectory problem is an initial value problemd y/dt=f(y),y(0)= where the interest lies in obtaining the curve traced by the solution (the trajectory), rather than in finding the actual correspondanc between values of the parametert and points on that curve. We prove the convergence of the Lambert-McLeod scheme for the numerical integration of trajectory problems. We also study the CELF method, an explicit procedure for the integration in time of semidiscretizations of PDEs which has some useful conservation properties. The proofs rely on the concept of restricted stability introduced by Stetter. In order to show the convergence of the methods, an idea of Strang is also employed, whereby the numerical solution is compared with a suitable perturbation of the theoretical solution, rather than with the theoretical solution itself.     

Nonuniqueness of solutions of Riemann problems

We investigate a general mechanism, utilizing nonclassical shock waves, for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two statesU ₁ andU ₂ such that a traveling wave leads fromU ₁ toU ₂ and another leads fromU ₂ toU ₁. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solutions of the Riemann problem, and we determine all solutions within a certain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of solutions of the conservation laws, as augmented by viscosity terms.     

Über eine Verallgemeinerung des Begriffs „Charakteristik”︁ bei partiellen Differentialgleichungen erster Ordnung

As, in general, the projections of characteristics into the x-space intersect for finite values of t, the global solution of a conservation law cannot be determined from the characteristic system of the equation, is considered. Only in the linear case, this equation coincides with the equation of the projections of characteristics. For convex h and all x₀ this equation has a solution almost everywhere, and the properties of this solution permit to construct a global solution of the conservation law using strips, in the same way as this is done for linear problems by the method of characteristics.     

A direct heuristic algorithm for linear programming

AnO(n ³) mathematically non-iterative heuristic procedure that needs no artificial variable is presented for solving linear programming problems. An optimality test is included. Numerical experiments depict the utility/scope of such a procedure.     

Convergence of a large time step generalization of Godunov's method for conservation laws

A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

A kinetic method for strictly nonlinear conservation laws

Ideas from kinetic theory are used to construct a new solution method for nonlinear conservation laws of the formu ₁+f(u)_x=0. We choose a class of distribution functionsG=G(t, x, ), which are compactly supported with respect to the artificial velocity. This can be done in an optimal way, i.e. so that the-integral of the solution of the linear kinetic equationG _t+G_x=0 solves the nonlinear conservation law exactly.Introducing a time step and variousx-discretisations one easily obtains a variety of numerical schemes. Among them are interesting new methods as well as known upstream schemes, which get a new interpretation and the possibility to incorporate boundary value problems this way.     

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

We consider partial differential equations of drift‐diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure‐valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0⁺. The time evolution of these singular masses is also explicitly described and, as a by‐product, uniqueness of this measure solution is obtained.     

A second order explicit finite element scheme to multidimensional conservation laws and its convergence

A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation laws.L ^p convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently smooth solutions a second order error estimate inL ² is proved under a stronger condition, Δt≤Ch ^2/4     

Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods

The approximate preservation of quadratic first integrals (QFIs) of differential systems in the numerical integration with Runge–Kutta (RK) methods is studied. Conditions on the coefficients of the RK method to preserve all QFIs up to a given order are obtained, showing that the pseudo-symplectic methods studied by Aubry and Chartier (BIT 98(3):439–461, 1998) of algebraic order p preserve QFIs with order q = 2p. An expression of the error of conservation of QFIs by a RK method is given, and a new explicit six-stage formula with classical order four and seventh order of QFI-conservation is obtained by choosing their coefficients so that they minimize both local truncation and conservation errors. Several formulas with algebraic orders 3 and 4 and different orders of conservation have been tested with some problems with quadratic and general first integrals. It is shown that the new fourth-order explicit method preserves much better the qualitative properties of the flow than the standard fourth-order RK method at the price of two extra function evaluations per step and it is a practical and efficient alternative to the fully implicit methods required for a complete preservation of QFIs.