We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton-Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720-742].The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications.At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes.The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton-Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier-Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

Abstract. A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36, 397-425, 2002]. Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously.Here, we introduce an improved second-order central-upwind scheme which, unlike its forerunners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one-and two-dimensional examples.

The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic partial differential equations. Abstractly, it can be viewed as determining a "good" n dimensional space H n to be used in approximating the elements of a compact set F in a Hilbert space H. One, by now popular, computational approach is to find H n through a greedy strategy. It is natural to compare the approximation performance of the H n generated by this strategy with that of the Kolmogorov widths d n (F) since the latter gives the smallest error that can be achieved by subspaces of fixed dimension n. The first such comparisons, given in [1], show that the approximation error, σ n (F) := dist(F, H n ), obtained by the greedy strategy satisfies σ n (F) ≤ Cn2 n d n (F). In this paper, various improvements of this result will be given. Among these, it is shown that whenever d n (F) ≤ M n −α , for all n > 0, and some M, α > 0, we also have σ n (F) ≤ C α M n −α for all n > 0, where C α depends only on α. Similar results are derived for generalized exponential rates of the form M e −an α . The exact greedy algorithm is not always computationally feasible and a commonly used computationally friendly variant can be formulated as a "weak greedy algorithm". The results of this paper are established for this version as well.

Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n dimensional space X n ⊂ X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width d n (F) X . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in [2] in the case X = H is a Hilbert space. The results of [2] were significantly improved on in [1]. The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.

This paper considers the problem of optimal recovery of an element u of a Hilbert space H from measurements of the form ℓ j (u), j = 1, . . . , m, where the ℓ j are known linear functionals on H. Problems of this type are well studied [18] and usually are carried out under an assumption that u belongs to a prescribed model class, typically a known compact subset of H. Motivated by reduced modeling for solving parametric partial differential equations, this paper considers another setting where the additional information about u is in the form of how well u can be approximated by a certain known subspace V n of H of dimension n, or more generally, in the form of how well u can be approximated by each of a sequence of nested subspaces V 0 ⊂ V 1 · · · ⊂ V n with each V k of dimension k. A recovery algorithm for the one-space formulation was proposed in [16]. Their algorithm is proven, in the present paper, to be optimal. It is also shown how the recovery problem for the one-space problem, has a simple formulation, if certain favorable bases are chosen to represent V n and the measurements. The major contribution of the present paper is to analyze the multi-space case. It is shown that, in this multi-space case, the set of all u that satisfy the given information can be described as the intersection of a family of known ellipsoids in H. It follows that a near optimal recovery algorithm in the multi-space problem is provided by identifying any point in this intersection. It is easy to see that the accuracy of recovery of u in the multi-space setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for u.