Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion

In this paper, we study the time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order 1+β∈[1,2] and the finite difference scheme for the time Caputo derivative with order α∈(0,1) (for subdiffusion) and (1,2) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0<α<1 is slower than that for α=1, but the diffusion velocity for 1<α<2 is faster than that for α=1. For the spatial diffusion, we have a similar observation.     

A set of compact finite-difference approximations to first and second derivatives,related to the extended Numerov method of Chawla on nonuniform grids

Summary The extended Numerov scheme of Chawla, adopted for nonuniform grids, is a useful compact finite-difference discretisation, suitable for the numerical solution of boundary value problems in singularly perturbed second order non-linear ordinary differential equations. A new set of three-point compact approximations to first and second derivatives, related to the Chawla scheme and valid for nonuniform grids, is developed in the present work. The approximations economically re-use intermediate quantities occurring in the Chawla scheme. The theoretical orders of accuracy are equal four for the central and one-sided first derivative approximations obtained, whereas the central second derivative formula is either fourth, third, or second order accurate, depending on the grid ratio. The approximations can be used for accurate a posteriori derivative evaluations. A Hermitian interpolation polynomial, consistent with the derivative approximations, is also derived. The values of the polynomial can be used, among other things, for guiding adaptive grid refinement. Accuracy orders of the new derivative approximations, and of the interpolating polynomial, are verified by computational experiments.      

Space-time Generalized Riemann Problem Solvers of Order k for Linear Advection with Unrestricted Time Step

This work concerns high-order approximations of the linear advection equation in very long time. A GRP-type scheme of arbitrary high-order in space and time with no restriction on the time step is developed. In the usual GRP solvers, we consider a polynomial approximation of the solution in space in each cell at the initial time. Here, we add a second polynomial approximation of the solution in time in each interface. Thanks to this double approximation, the resulting scheme is compact. It is proved to be of order k+1 in space and time, where k is the degree of the polynomials. Thanks to the compactness of the scheme, a two-dimensional extension is detailed on unstructured meshes made of triangles. Several numerical test-cases and comparison with existing methods illustrate the excellent behaviour of the scheme.     

On the Connection Between the Correction and Weighting Functions in the Correction Procedure via Reconstruction Method

In this note, the connection between the correction and weighting functions for the correction procedure via reconstruction (CPR) method in 1D is addressed. A one-parameter family of weighting functions is constructed from the discontinuous test space. It is found that if the solution polynomials lie in the space P ^k , then the first k weighting functions can always be chosen as the basis of the polynomial space P ^k?1 and the last weighting function can be selected as a piece-wise continuous polynomial. There exists at least one set of weighting functions which can recover the energy stable flux reconstruction (ESFR) schemes. This strategy has been successfully applied to recover several known high-order discontinuous schemes, including DG, SD, SV, and Huynh??s g ₂ scheme.     

The local discontinuous Galerkin method for 2D nonlinear time-fractional advection–diffusion equations

This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trapezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method.

     

A Comparison of HDG Methods for Stokes Flow

In this paper, we compare hybridizable discontinuous Galerkin (HDG) methods for numerically solving the velocity-pressure-gradient, velocity-pressure-stress, and velocity-pressure-vorticity formulations of Stokes flow. Although they are defined by using different formulations of the Stokes equations, the methods share several common features. First, they use polynomials of degree k for all the components of the approximate solution. Second, they have the same globally coupled variables, namely, the approximate trace of the velocity on the faces and the mean of the pressure on the elements. Third, they give rise to a matrix system of the same size, sparsity structure and similar condition number. As a result, they have the same computational complexity and storage requirement. And fourth, they can provide, by means of an element-by element postprocessing, a new approximation of the velocity which, unlike the original velocity, is divergence-free and H(div)-conforming. We present numerical results showing that each of the approximations provided by these three methods converge with the optimal order of k+1 in L ² for any k≥0. We also display experiments indicating that the postprocessed velocity is a better approximation than the original approximate velocity. It converges with an additional order than the original velocity for the gradient-based HDG, and with the same order for the vorticity-based HDG methods. For the stress-based HDG methods, it seems to converge with an additional order for even polynomial degree approximations. Finally, the numerical results indicate that the method based on the velocity-pressure-gradient formulation provides the best approximations for similar computational complexity.     

Numerical differentiation of implicitly defined space curves

In this note we develop a simple finite differencing device to calculate approximations of derivativesx′(0),x″(0),x ⁽³⁾(0), … of regular solution curvesx: ? ?s →x(s) ∈ ? ⁿ of nonlinear systems of equationsg(x)=0,g∈C ^k (? ^{n + 1}, ? ⁿ ) without having to compute points on the solution arcx(s). The derivative vectorsx′(0),x″(0),x ⁽³⁾(0),… can be used in the numerical approximation of the solution setg ^?1(0) in two ways. On one hand they can be applied to construct higher order predictors to be used in predictor-corrector branch following procedures. On the other they serve as order determining basis functions in the Reduced Basis Method. The performance of the differencing method is demonstrated by some numerical examples.     

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.     

A numerical approach to2k+e nonlinear equations with onlyk nonlinear variables

We derive a method for solving2k+e nonlinear algebraic equations in2k unknowns of the formL(x)y?b=0, which often occur in approximation theory. The method is based on solving a linear overdetermined system and a polynomial equation of thek-th order. To apply our method some numerical examples are presented.     

Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers’ Equation

This paper analyzes the stability and convergence of the Fourier pseudospectral method coupled with a variety of specially designed time-stepping methods of up to fourth order, for the numerical solution of a three dimensional viscous Burgers?? equation. There are three main features to this work. The first is a lemma which provides for an L ² and H ¹ bound on a nonlinear term of polynomial type, despite the presence of aliasing error. The second feature of this work is the development of stable time-stepping methods of up to fourth order for use with pseudospectral approximations of the three dimensional viscous Burgers?? equation. Finally, the main result in this work is that the pseudospectral method coupled with the carefully designed time-discretizations is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. It is notable that this stability condition does not impose a restriction on the time-step that is dependent on the spatial grid size, a fact that is especially useful for three dimensional simulations.     

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.     

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(h^k+1+τ^2?α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.     

On splitting-based numerical methods for nonlinear models of European options

We present a large class of nonlinear models of European options as parabolic equations with quasi-linear diffusion and fully nonlinear hyperbolic part. The main idea of the operator splitting method (OSM) is to couple known difference schemes for nonlinear hyperbolic equations with other ones for quasi-linear parabolic equations. We use flux limiter techniques, explicit–implicit difference schemes, Richardson extrapolation, etc. Theoretical analysis for illiquid market model is given. The numerical experiments show second-order accuracy for the numerical solution (the price) and Greeks Delta and Gamma, positivity and monotonicity preserving properties of the approximations.     

The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.     

Application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations

A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the C^∞ multiquadric (MQ) radial basis function (RBF) expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than 5 · 10⁻⁷.     

Stability properties of a higher order scheme for a GKdV-4 equation modelling surface water waves

This work is devoted to the study of a higher order numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p=4) in a bounded domain. The KdV equation and some of its generalizations as the GKdV type equations appear in Physics, for example in the study of waves on shallow water. Based on the analysis of stability of the first order scheme introduced by Pazoto et al. (Numer. Math. 116:317–356, 2010), we add a vanishing numerical viscosity term to a semi-discrete scheme so as to preserve similar properties of stability, and thus able to prove the convergence in L ⁴-strong. The semi-discretization of the spatial structure via central finite difference method yields a stiff system of ODE. Hence, for the temporal discretization, we resort to the two-stage implicit Runge-Kutta scheme of the Gauss-Legendre type. The resulting system is unconditionally stable and possesses favorable nonlinear properties. On the other hand, despite the formation of blow up for the critical case of GKdV, it is known that a localized damping term added to the GKdV-4 equation leads to the exponential decay of the energy for small enough initial conditions, which is interesting from the standpoint of the Control Theory. Then, combining the result of convergence in L ⁴-strong with discrete multipliers and a contradiction argument, we show that the presence of the vanishing numerical viscosity term allows the uniform (with respect to the mesh size) exponential decay of the total energy associated to the semi-discrete scheme of higher-order in space with the localized damping term. Numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results.     

A Hybrid Staggered Discontinuous Galerkin Method for KdV Equations

A hybrid staggered discontinuous Galerkin method is developed for the Korteweg–de Vries equation. The equation is written into a system of first order equations by introducing auxiliary variables. Two sets of finite element functions are introduced to approximate the solution and the auxiliary variables. The staggered continuity of the two finite element function spaces gives a natural flux condition and trace value on the element boundaries in the derivation of Galerkin approximation. On the other hand, to deal with the third order derivative term an hybridization idea is used and additional flux unknowns are introduced. The auxiliary variables can be eliminated in each element and the resulting algebraic system on the solution and the additional flux unknowns is solved. Stability of the semi discrete form is proven for various boundary conditions. Numerical results present the optimal order of \(L^2\)-errors of the proposed method for a given polynomial order.     

L ∞-error estimates of higher order mixed finite element approximations for elliptic optimal control problems

In this paper, we investigate the L ^∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ^∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems.     

Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes

In this article we propose the use of the ADER methodology of solving generalized Riemann problems to obtain a numerical flux, which is high order accurate in time, for being used in the Discontinuous Galerkin framework for hyperbolic conservation laws. This allows direct integration of the semi-discrete scheme in time and can be done for arbitrary order of accuracy in space and time. The resulting fully discrete scheme in time does not need more memory than an explicit first order Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme itself via the so-called Cauchy–Kovalewski procedure. We give an efficient algorithm for this procedure for the special case of the nonlinear two-dimensional Euler equations. Numerical convergence results for the nonlinear Euler equations results up to 8th order of accuracy in space and time are shown