Point-Value WENO Multiresolution Applications to Stable Image Compression

The ideas that lead from ENO to Weighted ENO (WENO) reconstructions (i.e. cell-average “interpolators”), devised and extensively used for the design of highly accurate shock capturing schemes for conservation laws, are applied in this paper to obtain weighted essentially non-oscillatory point-value nonlinear interpolators that can generically achieve an order of accuracy of 2r, when using stencils of 2r points at regions where the interpolated function is smooth. This interpolatory technique can be used in Harten’s multiresolution framework for image compression applications.     

A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

We analyze the spatial discretization errors associated with solutions of one-dimensional hyperbolic conservation laws by discontinuous Galerkin methods (DGMs) in space. We show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a Radau polynomial of degree p+1 on each element. We also prove that the local and global discretization errors are O(Δx^2(p+1)) and O(Δx^2p+1) at the downwind point of each element. This strong superconvergence enables us to show that local and global discretization errors converge as O(Δx^p+2) at the remaining roots of Radau polynomial of degree p+1 on each element. Convergence of local and global discretization errors to the Radau polynomial of degree p+1 also holds for smooth solutions as p→∞. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth.     

Concepts and Application of Time-Limiters to High Resolution Schemes

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.     

High Order Numerical Discretization for Hamilton–Jacobi Equations on Triangular Meshes

In this paper we construct several numerical approximations for first order Hamilton–Jacobi equations on triangular meshes. We show that, thanks to a filtering procedure, the high order versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are based on the first order Lax–Friedrichs scheme [2] which is improved here adjusting the dissipation term. The resulting first order scheme is -monotonic (we explain the expression in the paper) and converges to the viscosity solution as for the L -norm. The first high order method is directly inspired by the ENO philosophy in the sense where we use the monotonic Lax–Friedrichs Hamiltonian to reconstruct our numerical solutions. The second high order method combines a spatial high order discretization with the classical high order Runge–Kutta algorithm for the time discretization. Numerical experiments are performed for general Hamiltonians and L ¹, L ² and L -errors with convergence rates calculated in one and two space dimensions show the k-th order rate when piecewise polynomial of degree k functions are used, measured in L ¹-norm.     

Convex ENO Schemes for Hamilton–Jacobi Equations

In one dimension, viscosity solutions of Hamilton–Jacobi (HJ) equations can be thought as primitives of entropy solutions for conservation laws. Based on this idea, both theoretical and numerical concepts used for conservation laws can be passed to HJ equations even in several dimensions. In this paper, we construct convex ENO (CENO) schemes for HJ equations. This construction is a generalization from the work by Liu and Osher on CENO schemes for conservation laws. Several numerical experiments are performed. L ¹ and L ^∞ error and convergence rate are calculated as well.     

Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves

In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L²-norm is in both cases of optimal order and proportional to O(Δt²+h^p+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and Δt the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Non-linear filtering of counting processes driven by Ornstein-Uhlenbeck processes

The intensity of the doubly stochastic Poisson process (DSPP) considered in this paper is a linear function of a first-order Gauss-Markov process x ₁, (Ornstein-Uhlenbeck process).

By observing a DSPP realization and by analysing the conditional characteristic function of x ₁, we intend to find a non-linear recursive filter that gives an estimation of the intensity.

The expression of the centred conditional moments up to any order is established recursively, and a practical numerical algorithm is developed on the basis of a suboptimal non-linear filter. Consideration of the centred odd moments is also justified. The results of the numerical simulations are presented and enable a comparison to be made between the behaviour of the suboptimal non-linear filter and that of an adapted linear filter. The number of centred conditional moments to be retained in the formulation of the suboptimal non-linear filter is discussed.

Finally, numerical simulation results are given and commented on.     

ADER: Arbitrary High Order Godunov Approach

This paper concerns the construction of non-oscillatory schemes of very high order of accuracy in space and time, to solve non-linear hyperbolic conservation laws. The schemes result from extending the ADER approach, which is related to the ENO/WENO methodology. Our schemes are conservative, one-step, explicit and fully discrete, requiring only the computation of the inter-cell fluxes to advance the solution by a full time step; the schemes have optimal stability condition. To compute the intercell flux in one space dimension we solve a generalised Riemann problem by reducing it to the solution a sequence of m conventional Riemann problems for the kth spatial derivatives of the solution, with k=0, 1,..., m–1, where m is arbitrary and is the order of the accuracy of the resulting scheme. We provide numerical examples using schemes of up to fifth order of accuracy in both time and space.     

On Parabolic Boundary Layers for Convection–Diffusion Equations in a Channel: Analysis and Numerical Applications

In this article we discuss singularly perturbed convection–diffusion equations in a channel in cases producing parabolic boundary layers. It has been shown that one can improve the numerical resolution of singularly perturbed problems involving boundary layers, by incorporating the structure of the boundary layers into the finite element spaces, when this structure is available; see e.g. [Cheng, W. and Temam, R. (2002). Comput. Fluid. V.31, 453–466; Jung, C. (2005). Numer. Meth. Partial Differ. Eq. V.21, 623–648]. This approach is developed in this article for a convection–diffusion equation. Using an analytical approach, we first derive an approximate (simplified) form of the parabolic boundary layers (elements) for our problem; we then develop new numerical schemes using these boundary layer elements. The results are performed for the perturbation parameter ε in the range 10⁻¹–10⁻¹⁵ whereas the discretization mesh is in the range of order 1/10–1/100 in the x-direction and of order 1/10–1/30 in the y-direction. Indications on various extensions of this work are briefly described at the end of the Introduction.Dedicated to David Gottlieb on his 60th birthday.     

Frequency Optimized Computation Methods

In this paper we develop an alternative method to derive finite difference approximations of derivatives on arbitrary distrubutions of data points. The purpose is to find schemes which work for a broader range of frequencies than the usual approximations based on polynomial fitting to the expense of less accuracy for low frequencies. The numerical schemes are obtained as solutions to constrained optimizations problems in a weighted L ²-norm in the frequency domain. We examine the accuracy of these schemes and compare them with the standard approximations. To test the accuracy of the different schemes, we study dispersion errors for a simple wave equation in one space dimension. We examine the number of points per wave length which is needed in order for the relative error in the phase velocity to be below a certain bound. We also apply the technique to solve a simple two-dimensional hyperbolic equation.     

The numerical solution of fourth order mildly quasi-linear parabolic initial boundary value problem of second kind

In this article, we report on three-level implicit stable finite difference formulas of O(k ² + h ²) and O(k ² + h ⁴) for the numerical integration of certain mildly quasi-liner fourth order parabolic partial differential equations in one-space dimension. The numerical solution of u _xx is obtained as a by-product of the method. In all cases, we use only (3 + 3 + 3)-grid points and a single computational cell. Difference schemes for the fourth order linear parabolic equation in polar coordinates are also discussed. The stability analysis for the model linear problem is given as a representative example. Numerical results are presented to demonstrate the order and accuracy of the proposed methods.     

Analyzing a generalized Loop subdivision scheme

In this paper a class of subdivision schemes generalizing the algorithm of Loop is presented. The stencils have the same support as those from the algorithm of Loop, but allow a variety of weights. By varying the weights a class of C ¹ regular subdivision schemes is obtained. This class includes the algorithm of Loop and the midpoint schemes of order one and two for triangular nets. The proof of C ¹ regularity of the limit surface for arbitrary triangular nets is provided for any choice of feasible weights. The purpose of this generalization of the subdivision algorithm of Loop is to demonstrate the capabilities of the applied analysis technique. Since this class includes schemes that do not generalize box spline subdivision, the analysis of the characteristic map is done with a technique that does not need an explicit piecewise polynomial representation. This technique is computationally simple and can be used to analyze classes of subdivision schemes. It extends previously presented techniques based on geometric criteria.     

On an New Algorithm for Function Approximation with Full Accuracy in the Presence of Discontinuities Based on the Immersed Interface Method

This paper is devoted to the construction and analysis of an adapted and nonlinear multiresolution algorithm designed for interpolation or approximation of discontinuous univariate functions. The adaption attained allows to avoid numerical artifacts that appear when using linear algorithms and, at the same time, to obtain a high order of accuracy close to the singularities. It is known that linear algorithms are stable and convergent for smooth functions, but diffusion and Gibbs effect appear if the functions are piecewise continuous. Our aim is to develop an algorithm for function approximation with full accuracy that is capable to adapt to corners (kinks) and jump discontinuities, that uses a centered stencil and that does not use extrapolation. In order to reach this goal, we will need some information about the jumps in the function that we want to approximate and its derivatives. If this information is available, the algorithm is the most compact possible in the sense that the stencil is fixed and we do not need a stencil selection procedure as other algorithms do, such as ENO subcell resolution (ENO-SR). If the information about the jumps is not available, we will show a technique to approximate it. The algorithm is based on linear interpolation plus correction terms that provide the desired accuracy close to corners or jump discontinuities.     

Quadrature based finite element approximations to time dependent parabolic equations with nonsmooth initial data

The purpose of this paper is to study the effect of numerical quadrature in the finite element analysis for a time dependent parabolic equation with nonsmooth initial data. Both semidiscrete and fully discrete schemes are analyzed using standard energy techniques. For the semidiscrete case, optimal order error estimates are derived in the L ² and H ¹-norms and quasi-optimal order in the L ^∞-norm, when the initial function is only in H ₀ ¹. Finally, based on the backward Euler method, a time discretization scheme is discussed and almost optimal rates of convergence in the L ², H ¹ and L ^∞-norms are established. Received: September 1997 / Accepted: October 1997     

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.     

Eigenvalue problem for the Navier-Stokes operator in cylindrical coordinates

A fast, accurate, and robust numerical algorithm is proposed, suitable for parametric studies of incompressible fluid flow in a pipe. The new algorithm (or its fragments) can have a wider applicability, including cases when the computational domain contains a coordinate singularity along the polar axis r = 0 and when the dependence on the azimuth angle can be represented as a Fourier series, due to the physical symmetry of the problem. The constructed method enables the efficient solution of the eigenvalue problem for the linearized Navier-Stokes operator in cylindrical coordinates. The algorithm is based on a new change in the dependent variables, which makes it possible to circumvent the difficulties associated with coordinate singularities by taking into account the special behavior of analytic functions in the vicinity of the point r = 0. Despite the presence of coordinate singularities, the new algorithm ensures the spectral accuracy. The numerical solution of the linear problem of hydrodynamic stability involves the spatial discretization of the Navier-Stokes operator, its linearization about the stationary solution, and the reduction to the canonical eigenvalue problem of the type λx = Tx. Eigenvalues λ can then be calculated by the QR algorithm. An original method is proposed here for the reduction of the eigenvalue problem to its canonical form, employing the influence matrix technique. This method is economical and is characterized by its low sensitivity to round-off errors.     

Variational multiscale method based on the Crank–Nicolson extrapolation scheme for the non-stationary Navier–Stokes equations

In this report, a variational multiscale (VMS) method based on the Crank–Nicolson extrapolation scheme of time discretization for the turbulent flow is analysed. The flow is modelled by the fully evolutionary Navier–Stokes problem. This method has two differences compared to the standard VMS method: (i) For the trilinear term, we use the extrapolation skill to linearize the scheme; (ii) for the projection term, we lag it onto the previous time level to simplify the construction of the projection. These modifications make the algorithm more efficient and feasible. An unconditionally stability and an a priori error estimate are given for a case with rather general linear (cellwise constant) viscosity of the turbulent models. Moreover, numerical tests for both linear viscosity and nonlinear Smagorinsky-type viscosity are performed, they confirm the theoretical results and indicate the schemes are effective.     

Numerical solution of fully fuzzy linear systems by fuzzy neural network

In this paper, a new hybrid method based on fuzzy neural network (FNN) for approximate solution of fuzzy linear systems of the form Ax=d,Ax=d, where AA is a square matrix of fuzzy coefficients, xx and dd are fuzzy number vectors, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate solution of an n×nn\times n system of fuzzy linear equations that supposedly has a unique fuzzy solution, a simple algorithm from the cost function of the FNN is proposed. Finally, we illustrate our approach by some numerical examples.     

Superconvergence of Discontinuous Galerkin Methods for Convection-Diffusion Problems

Some discontinuous Galerkin methods for the linear convection-diffusion equation −ε u″+bu′=f are studied. Based on superconvergence properties of numerical fluxes at element nodes established in some earlier works, e.g., Celiker and Cockburn in Math. Comput. 76(257), 67–96, 2007, we identify superconvergence points for the approximations of u or q=u′. Our results are twofold: 1) For the minimal dissipation LDG method (we call it md-LDG in this paper) using polynomials of degree p, we prove that the leading terms of the discretization errors for u and q are proportional to the right Radau and left Radau polynomials of degree p+1, respectively. Consequently, the zeros of the right-Radau and left-Radau polynomials of degree p+1 are the superconvergence points of order p+2 for the discretization errors of the potential and of the gradient, respectively.