SUMMARYThe goal of the paper is to bring to the attention of the computational community a long overlooked, very simple, acceleration method that impressively speeds up explicit time-stepping schemes, at essentially no extra cost. The authors explain the basis of the method, namely stabilization via wisely chosen inner steps (stages), justify it for linear problems, and spell out how simple it is to incorporate in any explicit code for parabolic problems. Finally, we demonstrate its performance on the (linear) heat equation as well as on the (non-linear) classical Stefan problem, by comparing it with standard implicit schemes (employing SOR or Newton iterations). The results show that super-time-stepping is more efficient than the implicit schemes in that it runs at least as fast, it is of comparable or better accuracy, and it is, of course, much easier to program (and to parallelize for distributed computing).

A numerical model based on one-dimensional balance laws and ad hoc zero-dimensional boundary conditions is tested against experimental data. The study concentrates on the circle of Willis, a vital subnetwork of the cerebral vasculature. The main goal is to obtain efficient and reliable numerical tools with predictive capabilities. The flow is assumed to obey the Navier-Stokes equations, while the mechanical reactions of the arterial walls follow a viscoelastic model. Like many previous studies, a dimension reduction is performed through averaging. Unlike most previous work, the resulting model is both calibrated and validated against in vivo data, more precisely transcranial Doppler data of cerebral blood velocity. The network considered has three inflow vessels and six outflow vessels. Inflow conditions come from the data, while outflow conditions are modeled. Parameters in the outflow conditions are calibrated using a subset of the data through ensemble Kalman filtering techniques. The rest of the data is used for validation. The results demonstrate the viability of the proposed approach.

Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the µ(I), Φ(I)-rheology, which postulates that the bulk friction coefficient µ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction φ are functions of the inertial number I only. Although the µ(I), Φ(I)-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the µ(I), Φ(I)-rheology that does not suffer from such defects is proposed. In the framework of compressible I-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established µ(I) and Φ(I) relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

In this paper, new a postefiori error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least h 1/8 and h 1/4, respectively, in the LOO(L 1)-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.Key words, error estimates, streamline diffusion method, discontinuous galerkin method, multidimensional conservation laws AMS subject classifications. 65M60, 65N30, 35L65 Recently, Jaffr6, Johnson, and Szepessy [4] proved that the SCDG method converges to the entropy solution of problem (1.1), (1.2) by using an extension of the measure-value convergence theory of DiPerna [2] obtained by Szepessy 12] which allows them to replace in (i) the Loo((0, Too) x d)-norm by the Loo(0, Too; L2(d))-norm. In this paper, we consider the *Downloaded 11/28/14 to 129.120.242.61. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ERROR ESTIMATES FOR CONSERVATION LAWS 523 case of piecewise polynomial approximations of degree k and show how to obtain not only convergence, but also error estimates with only a suitable version of (ii); the properties (i) and (iii) do not need to be obtained. The basic idea is to combine the estimates of the entropy dissipation, needed in (ii), with a modification of the Kuznetsov approximation theory [8]; see also Cockbum, Coquel, and LeFloch 1]. The paper is organized as follows, in 2, we display the SCSD and SCDG methods and state and briefly discuss our main results. The remainder of the paper is devoted to proving them. In 3, we display a basic approximation inequality, Lemma 3.1. This inequality states that to obtain error estimates, only an estimate of the entropy dissipation is required. Those estimates are obtained in 4 and the a posteriori error estimates are proven. In 5, a very simple key regularity property of the approximate solution is obtained (by a simple L2-energy argument) which is then used to prove the remaining main results. In 6, we give a proof of Lemma 3.1 and we end in 7 with some concluding remarks.2. The main results. In this section, we describe the SCSD and SCDG finite element methods and state and briefly discuss our main results.The methods we have in mind being essentially implicit, we first decompose our domain into "slabs." More precisely, let 0 to < tl < t2 < < tN Too be a sequence of time levels. We set Sn (tn, tn+l) X d, n=0 N-l, d n tn} x ]I d, n O, N.

Abstract. In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for the Engquist-Osher scheme without using the fact (i) that the solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that the discrete semigroup associated with the scheme has the L 1 -contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.