We present methods for the reduction of the complexity of computational problems, both time-dependent and stationary, together with connections to renormalization, scaling, and irreversible statistical mechanics. Most of the methods have been presented before; what is new here is the common framework which relates the several constructions to each other and to methods of theoretical physics, as well as the analysis of the approximate reductions for time-dependent problems. The key conclusions are: (i) in time dependent problems, it is not in general legitimate to average equations without taking into account memory effects and noise; (ii) mathematical tools developed in physics for carrying out renormalization group transformations yield effective block Monte Carlo methods; (iii) the Mori–Zwanzig formalism, which in principle yields exact reduction methods but is often hard to use, can be tamed by approximation; and (iv) more generally, problem reduction is a search for hidden similarities.

The speed of propagation of a premixed turbulent flame correlates with the intensity of the turbulence encountered by the flame. One consequence of this property is that premixed flames in both laboratory experiments and practical combustors require some type of stabilization mechanism to prevent blow-off and flashback. The stabilization devices often introduce a level of geometric complexity that is prohibitive for detailed computational studies of turbulent flame dynamics. Furthermore, the stabilization introduces additional fluid mechanical complexity into the overall combustion process that can complicate the analysis of fundamental flame properties. To circumvent these difficulties we introduce a simple, heuristic feedback control algorithm that allows us to computationally stabilize a turbulent premixed flame in a simple geometric configuration. For the simulations, we specify turbulent inflow conditions and dynamically adjust the integrated fueling rate to control the mean location of the flame in the domain. We outline the numerical procedure, and illustrate the behavior of the control algorithm on methane flames at various equivalence ratios in two dimensions. The simulation data are used to study the local variation in the speed of propagation due to flame surface curvature.

This paper discusses the evaluation of hydrodynamic variables in the presence of spontaneous fluctuations, such as in molecular simulations of fluid flows. The principal point is that hydrodynamic variables such as fluid velocity and temperature must be defined in terms of mechanical variables such as momentum and energy density). Because these relations are nonlinear and because fluctuations of mechanical variables are correlated, care must be taken to avoid introducing a bias when evaluating means, variances, and correlations of hydrodynamic variables. The unbiased estimates are formulated; some alternative, incorrect approaches are presented as cautionary warnings. The expressions are verified by numerical simulations, both at thermodynamic equilibrium and at a nonequilibrium steady state.

The magnetohydrodynamic variational principle is employed to calculate equilibrium and stability of toroidal plasmas without two-dimensional symmetry. Differential equations are solved in a conservation form that describes force balance correctly across islands that are treated as discontinuities. The method is applied to both stellarators and tokamaks, and comparison with observations is favorable in both cases. Sometimes the solution of the equations turns out not to be unique, and there exist bifurcated equilibria that are nonlinearly stable when other theories predict linear instability. The calculations are consistent with recent measurements of high values of the pressure in stellarators. For tokamaks we compute three-dimensionally asymmetric solutions that are subject to axially symmetric boundary conditions.

In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and by modifying the operator if necessary. It has long been observed that the solution can be computed with uniform $O\left(h^2\right)$ accuracy even if the truncation error is $O\left(h\right)$ at the interface, while $O\left(h^2\right)$ in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to $O\left(h^{2}log\left(1∕h\right)\right)$. Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Two-fluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is $O\left(h^2\right)$.

A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an algorithm which computes the convolution of arbitrarily spaced data with the sinc kernel in $O\left(NlogN\right)$ operations, where $N$ denotes the number of data points. We briefly discuss its application to the construction of optimal density compensation weights for Fourier reconstruction and to the iterative approximation of the pseudoinverse of the signal equation in MRI.

We observe that, under very mild conditions, an $n$-dimensional space of functions (with a finite $n$) admits numerically stable $n$-point interpolation and integration formulae. The proof relies entirely on linear algebra, and is virtually independent of the domain and of the functions to be interpolated.

We investigate a new model for fluid flows in diatomite formations recently introduced by Barenblatt, Patzek, Prostokishin and Silin. We provide numerical evidences of the existence of a sharp front between the damaged and the undamaged rock, and we study the structure of this front. Finally, we simulate some infield operations and we set up a qualitative model control problem to maximize the profit during the oil extraction.

Spectral deferred correction is a flexible technique for constructing high-order, stiffly-stable time integrators using a low order method as a base scheme. Here we examine their use in conjunction with splitting methods to solve initial-boundary value problems for partial differential equations. We exploit their close connection with implicit Runge–Kutta methods to prove that up to the full accuracy of the underlying quadrature rule is attainable. We also examine experimentally the stability properties of the methods for various splittings of advection-diffusion and reaction-diffusion equations.

We compare the Immersed Interface Method (IIM) with the Extended Finite Element Method (X-FEM) for elliptic equations with singular sources and discontinuous coefficients. The IIM has been compared favorably with a number of other competing methods. These methods are of particular interest because they allow for the solution of elliptic equations with internal boundaries on nonconforming meshes. In the context of moving interface problems, the emphasis in this paper is placed on accuracy of solutions and their normal derivatives on the interface. These methods are briefly described and the results for benchmark problems are compared.