Fundamental Systems of Numerical Schemes for Linear Convection–Diffusion Equations and their Relationship to Accuracy

A new approach towards the assessment and derivation of numerical methods for convection dominated problems is presented, based on the comparison of the fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci. Received December 10, 1999; revised August 14, 2000     

On the Convergence of a Greedy Rank-One Update Algorithm for a Class of Linear Systems

In this paper we study the convergence of the well-known Greedy Rank-One Update Algorithm. It is used to construct the rank-one series solution for full-rank linear systems. The existence of the rank one approximations is also not new, but surprisingly the focus there has been more on the applications side more that in the convergence analysis. Our main contribution is to prove the convergence of the algorithm and also we study the required rank one approximation in each step. We also give some numerical examples and describe its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases. We illustrate this situation taking as a model problem the multidimensional Poisson equation with homogeneous Dirichlet boundary condition.     

Numerical Schemes of Diffusion Asymptotics and Moment Closures for Kinetic Equations

We investigate different models that are intended to describe the small mean free path regime of a kinetic equation, a particular attention being paid to the moment closure by entropy minimization. We introduce a specific asymptotic-induced numerical strategy which is able to treat the stiff terms of the asymptotic diffusive regime. We evaluate on numerics the performances of the method and the abilities of the reduced models to capture the main features of the full kinetic equation.     

An Asymptotic Preserving Method for Linear Systems of Balance Laws Based on Galerkin’s Method

We apply the concept of asymptotic preserving schemes (SIAM J Sci Comput 21:441–454, 1999) to the linearized \(p\) -system and discretize the resulting elliptic equation using standard continuous Finite Elements instead of Finite Differences. The fully discrete method is analyzed with respect to consistency, and we compare it numerically with more traditional methods such as Implicit Euler’s method.     

Numerical and Statistical Methods for the Coarse-Graining of Many-Particle Stochastic Systems

In this article we discuss recent work on coarse-graining methods for microscopic stochastic lattice systems. We emphasize the numerical analysis of the schemes, focusing on error quantification as well as on the construction of improved algorithms capable of operating in wider parameter regimes. We also discuss adaptive coarse-graining schemes which have the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. The methods employed in the development and the analysis of the algorithms rely on a combination of statistical mechanics methods (renormalization and cluster expansions), statistical tools (reconstruction and importance sampling) and PDE-inspired analysis (a posteriori estimates). We also discuss the connections and extensions of our work on lattice systems to the coarse-graining of polymers.     

Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for?Conservation Laws on Triangular Meshes

In Zhang and Shu (J. Comput. Phys. 229:3091–3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918–8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.     

On Convergence of the Arrow–Hurwicz Method for Saddle Point Problems

Journal of Mathematical Imaging and Vision - The Arrow–Hurwicz method is an inexact version of the Uzawa method; it has been widely applied to solve various saddle point problems in different...     

Stability and Convergence of Modified Du Fort–Frankel Schemes for Solving Time-Fractional Subdiffusion Equations

A class of modified Du Fort–Frankel-type schemes is investigated for fractional subdiffusion equations in the Jumarie’s modified Riemann–Liouville form with constant, variable or distributed fractional order. New explicit difference methods are constructed by combining the \(L1\) approximation of the modified fractional derivative with the idea of Du Fort–Frankel scheme, well-known for ordinary diffusion equations. Unconditional stability of the explicit methods is established in the sense of a discrete energy norm. The proposed schemes are shown to be convergent under the time-step (consistency) restriction of the classical Du Fort–Frankel scheme. Numerical examples are included to support our theoretical results.     

On the combination of logical and probabilistic models for?information analysis

Formal logical tools are able to provide some amount of reasoning support for information analysis, but are unable to represent uncertainty. Bayesian network tools represent probabilistic and causal information, but in the worst case scale as poorly as some formal logical systems and require specialized expertise to use effectively. We describe a framework for systems that incorporate the advantages of both Bayesian and logical systems. We define a formalism for the conversion of automatically generated natural deduction proof trees into Bayesian networks. We then demonstrate that the merging of such networks with domain-specific causal models forms a consistent Bayesian network with correct values for the formulas derived in the proof. In particular, we show that hard evidential updates in which the premises of a proof are found to be true force the conclusions of the proof to be true with probability one, regardless of any dependencies and prior probability values assumed for the causal model. We provide several examples that demonstrate the generality of the natural deduction system by using inference schemes not supportable directly in Horn clause logic. We compare our approach to other ones, including some that use non-standard logics.     

An Implementation of Haar Wavelet Based Method for Numerical Treatment of Time-fractional Schrodinger and Coupled Schr?dinger Systems

The objective of this paper is to solve the timefractional Schr?dinger and coupled Schr?dinger differential equations (TFSE) with appropriate initial conditions by using the Haar wavelet approximation. For the most part, this endeavor is made to enlarge the pertinence of the Haar wavelet method to solve a coupled system of time-fractional partial differential equations. As a general rule, piecewise constant approximation of a function at different resolutions is presentational characteristic of Haar wavelet method through which it converts the differential equation into the Sylvester equation that can be further simplified easily. Study of the TFSE is theoretical and experimental research and it also helps in the development of automation science, physics, and engineering as well. Illustratively, several test problems are discussed to draw an effective conclusion, supported by the graphical and tabulated results of included examples, to reveal the proficiency and adaptability of the method.      

On System Functions with the Property of Separability†

This paper considers linear differential (time-varying) systems which may be described by either of two system functions based on a specified integral transform. In particular, those systems are discussed for which at least one of the aforementioned system functions is separable in its two arguments. Physical interpretations of separable system functions are given and two theorems are proved which yield sufficient conditions for the presence of this property. It is also proved that the so-called ‘bi-frequency’ function of Zadeh must be Separable for linear differential systems. Finally, the problem of approximately representing a given system by a separable system function based on the Laplace transform is discussed.     

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in (Castro et al., Math. Comput. 75:1103–1134, 2006) to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems.     

Convergence Analysis in the Maximum Norm of the Numerical Gradient of the Shortley–Weller Method

The Shortley–Weller method is a standard central finite-difference-method for solving the Poisson equation in irregular domains with Dirichlet boundary conditions. It is well known that the Shortley–Weller method produces second-order accurate solutions and it has been numerically observed that the solution gradients are also second-order accurate; a property known as super-convergence. The super-convergence was proved in the \(L^{2}\) norm in Yoon and Min (J Sci Comput 67(2):602–617, 2016). In this article, we present a proof for the super-convergence in the \(L^{\infty }\) norm.     

ADER Schemes for Nonlinear Systems of Stiff Advection�CDiffusion�CReaction Equations

In this article we extend the high order ADER finite volume schemes introduced for stiff hyperbolic balance laws by Dumbser, Enaux and Toro (J. Comput. Phys. 227:3971?C4001, 2008) to nonlinear systems of advection?Cdiffusion?Creaction equations with stiff algebraic source terms. We derive a new efficient formulation of the local space-time discontinuous Galerkin predictor using a nodal approach whose interpolation points are tensor-products of Gauss?CLegendre quadrature points. Furthermore, we propose a new simple and efficient strategy to compute the initial guess of the locally implicit space-time DG scheme: the Gauss?CLegendre points are initialized sequentially in time by a second order accurate MUSCL-type approach for the flux term combined with a Crank?CNicholson method for the stiff source terms. We provide numerical evidence that when starting with this initial guess, the final iterative scheme for the solution of the nonlinear algebraic equations of the local space-time DG predictor method becomes more efficient. We apply our new numerical method to some systems of advection?Cdiffusion?Creaction equations with particular emphasis on the asymptotic preserving property for linear model systems and the compressible Navier?CStokes equations with chemical reactions.     

On the C-property and Generalized C-property of?Residual Distribution for the Shallow Water Equations

In this paper we consider the discretization of the Shallow Water equations by means of Residual Distribution (RD) schemes. We review the conditions allowing the exact preservation of some exact steady solutions. These conditions are shown to be related both to the type of spatial approximation and to the quadrature used to evaluate the cell residual. Numerical examples are shown to validate the theory.     

On the Numerical Controllability of the Two-Dimensional Heat,Stokes and Navier–Stokes Equations

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier–Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov–Imanuvilov formulation, see Fursikov and Imanuvilov (Controllability of Evolutions Equations, Lectures Notes Series, vol 34, Seoul National University, 1996); this approach has been followed recently for the one-dimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables \(x_1\), \(x_2\) and t that is second-order in time and fourth-order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments.     

On the Convergence of Primal–Dual Hybrid Gradient Algorithms for Total Variation Image Restoration

In this paper we establish the convergence of a general primal?Cdual method for nonsmooth convex optimization problems whose structure is typical in the imaging framework, as, for example, in the Total Variation image restoration problems. When the steplength parameters are a priori selected sequences, the convergence of the scheme is proved by showing that it can be considered as an ??-subgradient method on the primal formulation of the variational problem. Our scheme includes as special case the method recently proposed by Zhu and Chan for Total Variation image restoration from data degraded by Gaussian noise. Furthermore, the convergence hypotheses enable us to apply the same scheme also to other restoration problems, as the denoising and deblurring of images corrupted by Poisson noise, where the data fidelity function is defined as the generalized Kullback?CLeibler divergence or the edge preserving removal of impulse noise. The numerical experience shows that the proposed scheme with a suitable choice of the steplength sequences performs well with respect to state-of-the-art methods, especially for Poisson denoising problems, and it exhibits fast initial and asymptotic convergence.     

On the Numerical Approximation of the Length of (Implicit) Level Curves

The evaluation of the length of a curve, represented in an Eulerian way as the zero level set of an implicit function, depends mainly on the representation of the curve. In this paper, we propose a parameter to measure the complexity of the curve, and therefore the accuracy of the evaluation, based on the evolution of the representation in different scales. We will analyze this parameter, its properties and its relations with the regularity of the curve.     

On Controllability and Observability of Multivariable Linear Time-invariant Systems

Some results on linear system theory are reported.Based on these results,necessary and sufficient conditions for the controllability and observability of both continuous-time and its corresponding discrete-time multivariable linear time-invariant systems are presented.