High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.     

High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order \(O(\tau ^{3-\alpha } + h^2), 0<\alpha <1\) are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699–A2724, 2016), where \(\tau \) and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.     

Well Conditioned Pseudospectral Schemes with Tunable Basis for Fractional Delay Differential Equations

The main purpose of this work is to develop spectrally accurate and well conditioned pseudospectral schemes for solving fractional delay differential equations (FDDEs). The essential idea is to recast FDDEs into fractional integral equations (FIEs) and then discretize the FIEs via generalized fractional pseudospectral integration matrices (GFPIMs). We construct GFPIMs by employing the basis of weighted Lagrange interpolating functions, and provide an exact, efficient, and stable approach to computing GFPIMs. The GFPIM schemes have two remarkable features: (i) the endpoint singularity of the solution to FDDEs can be effectively captured via the tunable basis, and (ii) the linear system resulting from pseudospectral discretization is well conditioned. We also provide a rigorous convergence analysis for the particular FPIM schemes via a linear FIE with any \(\gamma >0\) where \(\gamma \) is the order of fractional integrals. Numerical results on benchmark FDDEs with smooth/singular solutions demonstrate the spectral rate of convergence for the GFPIM schemes. For FDDEs with piecewise smooth solutions, the GFPIM schemes can obtain accurate solutions but converge slowly due to their essential feature of “global” approximation on the entire time interval.     

Arbitrary High-Order Discontinuous Galerkin Schemes for the Magnetohydrodynamic Equations

In this paper, we propose a discontinuous Galerkin scheme with arbitrary order of accuracy in space and time for the magnetohydrodynamic equations. It is based on the Arbitrary order using DERivatives (ADER) methodology: the high order time approximation is obtained by a Taylor expansion in time. In this expansion all the time derivatives are replaced by space derivatives via the Cauchy-Kovalevskaya procedure. We propose an efficient algorithm of the Cauchy-Kovalevskaya procedure in the case of the three-dimensional magneto-hydrodynamic (MHD) equations. Parallel to the time derivatives of the conservative variables the time derivatives of the fluxes are calculated. This enables the analytic time integration of the volume integral as well as that of the surface integral of the fluxes through the grid cell interfaces which occur in the discrete equations. At the cell interfaces the fluxes and all their derivatives may jump. Following the finite volume ADER approach the break up of all these jumps into the different waves are taken into account to get proper values of the fluxes at the grid cell interfaces. The approach under considerations is directly based on the expansion of the flux in time in which the leading order term may be any numerical flux calculation for the MHD-equation. Numerical convergence results for these equations up to 7th order of accuracy in space and time are shown.     

Quasi-Compact Finite Difference Schemes for Space Fractional Diffusion Equations

In this paper, a compact difference operator, termed CWSGD, is designed to establish the quasi-compact finite difference schemes for approximating the space fractional diffusion equations in one and two dimensions. The method improves the spatial accuracy order of the weighted and shifted Grünwald difference (WSGD) scheme (Tian et al., arXiv:1201.5949) from 2 to 3. The numerical stability and convergence with respect to the discrete L ² norm are theoretically analyzed. Numerical examples illustrate the effectiveness of the quasi-compact schemes and confirm the theoretical estimations.     

Spectral Methods for Substantial Fractional Differential Equations

In this paper, a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial fractional differential equations are proposed, analyzed, and tested. We modify a class of generalized Laguerre polynomials to form our trial basis and test basis. After a proper scaling of these bases, our Petrov–Galerkin method results in diagonal and well-conditioned linear systems for certain types of fractional differential equations. In the meantime, we provide superconvergence points of the Petrov–Galerkin approximation for associated fractional derivative and function value of true solution. Additionally, we present explicit fractional differential collocation matrices based upon Laguerre–Gauss–Radau points. It is noteworthy that the proposed methods allow us to adjust a parameter in the basis according to different given data to maximize the convergence rate. All these findings have been proved rigorously in our convergence analysis and confirmed in our numerical experiments.     

A High-Order Accurate Method for Frequency Domain Maxwell Equations with Discontinuous Coefficients

Maxwell equations contain a dielectric coefficient ɛ that describes the particular media. For homogeneous materials the dielectric coefficient is constant. There is a jump in this coefficient across the interface between differing media. This discontinuity can significantly reduce the order of accuracy of the numerical scheme. We present an analysis and implementation of a fourth order accurate algorithm for the solution of Maxwell equations with an interface between two media and so the dielectric coefficient is discontinuous. We approximate the discontinuous function by a continuous one either locally or in the entire domain. We study the one-dimensional system in frequency space. We only consider schemes that can be implemented for multidimensional problems both in the frequency and time domains.     

Supervised Local High-Order Differential Channel Feature Learning for Pedestrian Detection

In this paper, a novel supervised local high-order differential channel feature is proposed for fast pedestrian detection. This method is motivated by the recent successful use of filtering on the multiple channel maps, which can improve the performance. This method firstly compute the multiple channel maps for the input RGB image, and average pooling is acted on the channel maps in order to reduce the effect of noise and sample misalignment. Then, each of the pooled channel maps is convolved with our proposed local high-order filter bank, which can enhance the discriminative information in the feature space. Finally, due to the increasing memory consumption incurred by the higher dimension of resulting feature, we have proposed a local structure preserved supervised dimension reduction method which aims to keep the manifold structure of samples in the feature space. This method is formulated as a classical spectral graph embedding problem which can be solved by the LPP algorithms. Thorough experiments and comparative studies show that our method can achieve very competitive result compared with many state-of-art methods on the INRIA and Caltech datasets. Besides, our detector can run about 20 fps in 480 \(\times \) 640 resolution images.     

Error Inhibiting Block One-step Schemes for Ordinary Differential Equations

The commonly used one step methods and linear multi-step methods all have a global error that is of the same order as the local truncation error (as defined in [1, 6, 8, 13, 15]). In fact, this is true of the entire class of general linear methods. In practice, this means that the order of the method is typically defined solely by order conditions which are derived by studying the local truncation error. In this work we investigate the interplay between the local truncation error and the global error, and develop a methodology which defines the construction of explicit error inhibiting block one-step methods (alternatively written as explicit general linear methods [2]). These error inhibiting schemes are constructed so that the accumulation of the local truncation error over time is controlled, which results in a global error that is one order higher than the local truncation error. In this work, we delineate how to carefully choose the coefficient matrices so that the growth of the local truncation error is inhibited. We then use this theoretical understanding to construct several methods that have higher order global error than local truncation error, and demonstrate their enhanced order of accuracy on test cases. These methods demonstrate that the error inhibiting concept is realizable. Future work will further develop new error inhibiting methods and will analyze the computational efficiency and linear stability properties of these methods.     

High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order \(\mathcal{O}(\tau ^{r}+h^{4})\), where \(r\ge 2\) is a positive integer and \(\tau \) and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of \(2\le r\le 6\), and the optimal error estimates in the weighted \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.     

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations (FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.      

TVD Fluxes for the High-Order ADER Schemes

In this paper we propose to use a TVD flux, instead of a first-order monotone flux, as the building block for designing very high-order methods; we implement the idea in the context of ADER schemes via a new flux expansion. Systematic assessment of the new schemes shows substantial gains in accuracy; these are particularly evident for problems involving long time evolution     

Second-Order Accurate Computation of Curvatures in a Level Set Framework Using Novel High-Order Reinitialization Schemes

We present a high-order accurate scheme for the reinitialization equation of Sussman et al.(J. Comput. Phys. 114:146–159, [1994]) that guarantees accurate computation of the interface’s curvatures in the context of level set methods. This scheme is an extension of the work of Russo and Smereka (J. Comput. Phys. 163:51–67, [2000]). We present numerical results in two and three spatial dimensions to demonstrate fourth-order accuracy for the reinitialized level set function, third-order accuracy for the normals and second-order accuracy for the interface’s mean curvature in the L ¹- and L ^∞-norms. We also exploit the work of Min and Gibou (UCLA CAM Report (06-22), [2006]) to show second-order accurate scheme for the computation of the mean curvature on non-graded adaptive grids.     

A New Class of Highly Accurate Solvers for Ordinary Differential Equations

We introduce a new class of predictor-corrector schemes for the numerical solution of the Cauchy problem for non-stiff ordinary differential equations (ODEs), obtained via the decomposition of the solutions into combinations of appropriately chosen exponentials; historically, such techniques have been known as exponentially fitted methods. The proposed algorithms differ from the classical ones both in the selection of exponentials and in the design of the quadrature formulae used by the predictor-corrector process. The resulting schemes have the advantage of significantly faster convergence, given fixed lengths of predictor and corrector vectors. The performance of the approach is illustrated via a number of numerical examples. This work was partially supported by the US Department of Defense under ONR Grant #N00014-07-1-0711 and AFOSR Grants #FA9550-06-1-0197 and #FA9550-06-1-0239.     

Limited Memory Block Preconditioners for Fast Solution of Fractional Partial Differential Equations

An innovative block structured with sparse blocks multi iterative preconditioner for linear multistep formulas used in boundary value form is proposed here to accelerate GMRES, FGMRES and BiCGstab(l). The preconditioner is based on block \(\omega \)-circulant matrices and a short-memory approximation of the underlying Jacobian matrix of the fractional partial differential equations. Convergence results, numerical tests and comparisons with other techniques confirm the effectiveness of the approach.     

The Multi-scale Method for Solving Nonlinear Time Space Fractional Partial Differential Equations

In this paper, we present a new algorithm to solve a kind of nonlinear time space-fractional partial differential equations on a finite domain. The method is based on B-spline wavelets approximations, some of these functions are reshaped to satisfy on boundary conditions exactly. The Adams fractional method is used to reduce the problem to a system of equations. By multiscale method this system is divided into some smaller systems which have less computations. We get an approximated solution which is more accurate on some subdomains by combining the solutions of these systems. Illustrative examples are included to demonstrate the validity and applicability of our proposed technique, also the stability of the method is discussed.