Showing papers on "Finite difference published in 2013"

A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods

Abstract: This paper shows that the discontinuous Galerkin collocation spectral element method with Gauss--Lobatto points (DGSEM-GL) satisfies the discrete summation-by-parts (SBP) property and can thus be classified as an SBP-SAT (simultaneous approximation term) scheme with a diagonal norm operator. In the same way, SBP-SAT finite difference schemes can be interpreted as discontinuous Galerkin-type methods with a corresponding weak formulation based on an inner-product formulation common in the finite element community. This relation allows the use of matrix-vector notation (common in the SBP-SAT finite difference community) to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for DGSEM-GL and diagonal norm SBP-SAT. Based on this result, a skew-symmetric energy stable discretely conservative DGSEM-GL formulation (applicable to general diagonal norm SBP-SAT schemes) for the nonlinear Burgers equation is constructed.

The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation

Abstract: In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and convergent of order $O(\tau^q+h^{r+1})$ in the $L^2$ norm, where $q=2-\beta$ or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth, $\beta\,(0<\beta<1)$ is the order of the fractional derivative, $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the polynomial space. The corresponding schemes for the subdiffusion equation with Neumann boundary conditions are presented as well, where the stability and convergence are shown. Numerical examples are provided to verify the theoretical analysis. Comparisons between the algorithms derived in this paper and the existing algorithms are given, ...

Quantum calculus on finite intervals and applications to impulsive difference equations

Abstract: In this paper we initiate the study of quantum calculus on finite intervals. We define the qk-derivative and qk-integral of a function and prove their basic properties. As an application, we prove existence and uniqueness results for initial value problems for first- and second-order impulsive qk-difference equations. MSC: 26A33; 39A13; 34A37

Stable calculation of Gaussian-based RBF-FD stencils

Abstract: Traditional finite difference (FD) methods are designed to be exact for low degree polynomials. They can be highly effective on Cartesian-type grids, but may fail for unstructured node layouts. Radial basis function-generated finite difference (RBF-FD) methods overcome this problem and, as a result, provide a much improved geometric flexibility. The calculation of RBF-FD weights involves a shape parameter @e. Small values of @e (corresponding to near-flat RBFs) often lead to particularly accurate RBF-FD formulas. However, the most straightforward way to calculate the weights (RBF-Direct) becomes then numerically highly ill-conditioned. In contrast, the present algorithm remains numerically stable all the way into the @e->0 limit. Like the RBF-QR algorithm, it uses the idea of finding a numerically well-conditioned basis function set in the same function space as is spanned by the ill-conditioned near-flat original Gaussian RBFs. By exploiting some properties of the incomplete gamma function, it transpires that the change of basis can be achieved without dealing with any infinite expansions. Its strengths and weaknesses compared with the Contour-Pade, RBF-RA, and RBF-QR algorithms are discussed.

Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations

Abstract: Gradient schemes are nonconforming methods written in discrete variational formulation and based on independent approximations of functions and gradients, using the same degrees of freedom. Previous works showed that several well-known methods fall in the framework of gradient schemes. Four properties, namely coercivity, consistency, limit-conformity and compactness, are shown in this paper to be sufficient to prove the convergence of gradient schemes for linear and nonlinear elliptic and parabolic problems, including the case of nonlocal operators arising for example in image processing. We also show that the schemes of the Hybrid Mimetic Mixed family, which include in particular the Mimetic Finite Difference schemes, may be seen as gradient schemes meeting these four properties, and therefore converges for the class of above-mentioned problems.

Quasi-Compact Finite Difference Schemes for Space Fractional Diffusion Equations

Abstract: 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 Grunwald 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.

Globally optimal finite-difference schemes based on least squares

Abstract: Spatial finite-difference (FD) coefficients are usually determined by the Taylor-series expansion (TE) or optimization methods. The former can provide high accuracy on a smaller wavenumber or frequency zone, and the latter can give moderate accuracy on a larger zone. Present optimization methods applied to calculate FD coefficients are generally gradient-like or global optimization-like algorithms, and thus iterations are involved. They are more computationally expensive, and sometimes the global solution may not be found. I examined second-order spatial derivatives and computed the optimized spatial FD coefficients over the given wavenumber range using the least-squares (LS) method. The results indicated that the FD accuracy increased with increasing operator length and decreasing wavenumber range. Therefore, for the given error and operator length, globally optimal spatial FD coefficients can be easily obtained. Some optimal FD coefficients were given. I developed schemes to obtain optimized LS-...

Two finite difference schemes for time fractional diffusion-wave equation

Abstract: Time fractional diffusion-wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. In this paper we construct two finite difference schemes to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations. Under the weak smoothness conditions, we prove that our two schemes are convergent with first-order accuracy in temporal direction and second-order accuracy in spatial direction. Numerical experiments are carried out to demonstrate the theoretical analysis.

Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solution with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.

Optimized finite-difference operator for broadband seismic wave modeling

Abstract: High-resolution image and waveform inversion of smallscale targets requires the handling of high-frequency seismic wavefields. However, conventional finite-difference (FD) methods have strong numerical dispersions in the presence of high-frequency components. To reduce these numerical dispersions, we optimized the constant coefficients of the FD operator by maximizing the wavenumber coverage within a given error limitation. We set up three general criteria to enhance the convergence of the algorithm and reduce the optimization effort. We selected the error limitation to be 0.0001, this being the smallest in the literature, which led to perfect agreement between theoretical analyses and numerical experiments. The accuracy of our optimized FD methods can even reach that of much higher order unoptimized FD methods, which means great savings of computational efforts and memory demand. These advantages become even more apparent with 3D modeling, especially for saving memory demand.

Proposal for a Standard Micromagnetic Problem: Spin Wave Dispersion in a Magnonic Waveguide

Abstract: In this paper, we propose a standard micromagnetic problem, of a nanostripe of permalloy. We study the magnetization dynamics and describe methods of extracting features from simulations. Spin wave dispersion curves, relating frequency and wave vector, are obtained for wave propagation in different directions relative to the axis of the waveguide and the external applied field. Simulation results using both finite element (Nmag) and finite difference (OOMMF) methods are compared against analytic results, for different ranges of the wave vector.

Convergence of a Discontinuous Galerkin scheme for the mixed time domain Maxwell's equations in dispersive media.

Abstract: This work is about the numerical solution of the time-domain Maxwell's equations in dispersive propagation media by a discontinuous Galerkin time-domain method. The Debye model is used to describe the dispersive behaviour of the media. The resulting system of differential equations is solved using a centred-flux discontinuous Galerkin formulation for the discretization in space and a second-order leapfrog scheme for the integration in time. The numerical treatment of the dispersive model relies on an auxiliary differential equation approach similar to that which is adopted in the finite difference time-domain method. Stability estimates are derived through energy considerations and convergence is proved for both the semidiscrete and the fully discrete schemes.

Modeling of Complex Geometries and Boundary Conditions in Finite Difference/Finite Volume Time Domain Room Acoustics Simulation

Abstract: Due to recent increases in computing power, room acoustics simulation in 3D using time stepping schemes is becoming a viable alternative to standard methods based on ray tracing and the image source method. Finite Difference Time Domain (FDTD) methods, operating over regular grids, are perhaps the best known among such methods, which simulate the acoustic field in its entirety over the problem domain. In a realistic room acoustics setting, working over a regular grid is attractive from a computational standpoint, but is complicated by geometrical considerations, particularly when the geometry does not conform neatly to the grid, and those of boundary conditions which emulate the properties of real wall materials. Both such features may be dealt with through an appeal to methods operating over unstructured grids, such as finite volume methods, which reduce to FDTD when employed over regular grids. Through numerical energy analysis, such methods lead to direct stability conditions for complex problems, including convenient geometrical conditions at irregular boundaries. Simulation results are presented.

Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation

Abstract: High-order compact finite difference method with operator-splitting technique for solving the two dimensional time fractional diffusion equation is considered in this paper. The Caputo derivative is evaluated by the L1 approximation, and the second order derivatives with respect to the space variables are approximated by the compact finite differences to obtain fully discrete implicit schemes. Alternating Direction Implicit (ADI) method is used to split the original problem into two separate one dimensional problems. One scheme is given by replacing the unknowns by the values on the previous level directly and a correction term is added for another scheme. Theoretical analysis for the first scheme is discussed. The local truncation error is analyzed and the stability is proved by the Fourier method. Using the energy method, the convergence of the compact finite difference scheme is proved. Numerical results are provided to verify the accuracy and efficiency of the two proposed algorithms. For the order of the temporal derivative lies in different intervals $\left(0,\frac{1}{2}\right)$ or $\left[\frac{1}{2},1\right)$ , corresponding appropriate scheme is suggested.

Computational methods in the fractional calculus of variations and optimal control

Abstract: The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.

Convergent Filtered Schemes for the Monge--Ampère Partial Differential Equation

Abstract: The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear partial differential equations such as the elliptic Monge--Ampere equation. The approximation theory of Barles and Souganidis [Asymptotic Anal., 4 (1991), pp. 271--283] requires that numerical schemes be monotone (or elliptic in the sense of [A. M. Oberman, SIAM J. Numer. Anal., 44 (2006), pp. 879--895]). But such schemes have limited accuracy. In this article, we establish a convergence result for filtered schemes, which are nearly monotone. This allows us to construct finite difference discretizations of arbitrarily high-order. We demonstrate that the higher accuracy is achieved when solutions are sufficiently smooth. In addition, the filtered scheme provides a natural detection principle for singularities. We employ this framework to construct a formally second-order scheme for the Monge--Ampere equation and present computational results on smooth and singular solutions.