Diffusion-wave Equation Research Articles

Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.

Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: Meshless interpolating element free Galerkin (IEFG) and finite element methods

In this paper we apply a finite element scheme and interpolating element free Galerkin technique for the numerical solution of the two-dimensional time fractional diffusion-wave equation on the irregular domains. The time fractional derivative which has been described in the Caputo׳s sense is approximated by a scheme of order O(τ3−α), 1<α<2, and the space derivatives are discretized with finite element and interpolating element free Galerkin techniques. We prove the unconditional stability and obtain an error bound for the two new schemes using the energy method. However we would like to emphasize that the main aim of the current paper is to implement the Galerkin finite element method and interpolating element free Galerkin method on complex domains. Also we present error estimate for both schemes proposed for solving the time fractional diffusion-wave equation. Numerical examples demonstrate the theoretical results and the efficiency of the proposed scheme.

Numerical Solution of Fractional Diffusion-Wave Equation

ABSTRACTIn this article, a novel compact finite difference scheme is \\mboxconstructed to solve the fractional diffusion-wave equation based on its equivalent integro-differential equation. In the temporal direction, the product trapezoidal scheme is employed to treat the fractional integral term. The convergence and stability of the scheme are proved. Numerical examples are also provided to verify the theoretical analysis.

A Systematic Spectral-Tau Method for the Solution of Fuzzy Fractional Diffusion and Fuzzy Fractional Wave Equations

A Systematic Spectral-Tau Method for the Solution of Fuzzy Fractional Diffusion and Fuzzy Fractional Wave Equations

A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations

In this paper, we have considered the fractional diffusion and fractional diffusion-wave equations in which the time derivative is a fractional derivative in the Caputo form and have obtained their numerical solutions by collocation method using cubic B-spline base functions. In the solution process, for the fractional diffusion equation $L1$ discretizaton formula of the fractional derivative is applied, and for the fractional diffusion-wave equation $L2$ discretizaton formula of the fractional derivative is applied. Accuracy of the proposed method is discussed by computing the error norms $L_{2}$ and $L_{\infty}$ . A stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

A coupled stream flow and depth-integrated subsurface flow model for catchment hydrology

Few hydrological models that couple stream flow and subsurface flow in shallow aquifers are based on a compromise between simple and complex depiction of the system, although this compromise could result in tractable tools for various applications. We present a depth-integrated approach in which flows in the vadose and saturated zones are assumed to be parallel to the bottom of the aquifer and thus are integrated in the direction normal to the bottom of the aquifer. The hydrodynamic parameters are also integrated in this direction, and gravity effects are preserved. Stream flow is handled by a diffusive-wave equation that is calculated over a network of one-dimensional bonds. The first-order coupling between the stream and subsurface flows exchanges water fluxes between the stream network and the subsurface compartment according to the hydraulic head differences between the systems. Three synthetic test cases, one including a comparison with a three-dimensional code, are used to evaluate the general behavior of the coupled model. It is shown that the approach reproduces the main hydrological features at the catchment scale, including the generation of runoff, infiltration–exfiltration into (from) the vadose zone, and smooth transient head variations in the aquifer.

The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests

We present a distributed-order fractional diffusion-wave equation (dofDWE) to describe radial groundwater flow to or from a well, and three sets of solutions of the dofDWE for flow from a well for aquifer tests: one for pumping tests, and two for slug tests. The dofDWE is featured by two temporal orders of fractional derivatives, b1 and b2, which characterise small and large pores, respectively. By fitting the approximate solutions of the dofDWE to data from slug tests in the field, we determined the effective saturated hydraulic conductivity, Ke, transmissivity, Tf, and the order of fractional derivatives, b2 in one test and b2 and b1 in the second test. We found that the patterns of groundwater flow from a well during the slug tests at this site belong to the class of sub-diffusion with b2 < 1 and b1 < 1 using both the short-time and large-time solutions. We introduce the concept of the critical time to link Ke as a function of b2 and b1. The importance of the orders of fractional derivatives is obvious in the approximate solutions: for short time slug tests only the parameter b2 for flow in large pores is present while for long time slug tests the parameters b2 and b1 are present indicating both large and small pores are functioning.

Solution of nonlinear fractional diffusion-wave equation by traingular functions

In this paper, we propose new technique for solving fractional diffusion-wave equation. This algorithm is based on Triangular Function (TFs) methods, a new fractional operational matrix of integration for the TFs is derived. The TFs and their fractional operational matrix of integration are used to transform the problem under consideration into a linear system of algebraic equations. The new technique is applied to solving two mathematical models. We show that the present approach is relatively easy, efficient and highly accurate.

Four free surface algorithms for the 3D Navier–Stokes equations

Four algorithms are described for computing a steady free water surface with the solution of the three-dimensional (3D) Navier–Stokes equations. The numerical methods are used in hydraulic engineering cases, typically spillways and river modelling. The algorithms were tested against a laboratory experiment of a v-shaped broad-crested weir. The complex geometry of the weir introduced three-dimensional effects, which the numerical methods handled with varying degrees of success. One of the methods tested was the classical volume of fluid (VOF) approach, implemented in the OpenFOAM software with a fixed grid. The other three algorithms used an adaptive grid that followed the free water surface. These methods were coded in the SSIIM 2 program and were based on water continuity, pressure differences and an implicit solution of the diffusive wave equation. The VOF method gave the best results compared with the experiments. However, this method requires a very short time step. Two of the investigated methods compute the water surface location implicitly and can therefore use a much longer time step. The method based on the diffusive wave equation has the disadvantage that the results depend on a calibrated friction factor. All four methods predicted the water depth over the weir with an average accuracy below 14%.

Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains

In this paper, we derive and analyse a compact difference scheme for a distributed-order time-fractional diffusion-wave equation. This equation is approximated by a multi-term fractional diffusion-wave equation, which is then solved by a compact difference scheme. The unique solvability of the difference solution is discussed. Using the discrete energy method, we prove the compact difference scheme is unconditionally stable and convergent. Finally, numerical results are presented to support our theoretical analysis.

Matrix method based on the second kind Chebyshev polynomials for solving time fractional diffusion-wave equations

In this paper, the second kind Chebyshev polynomials (SKCPs) basis is used to solve time-fractional diffusion-wave equations with damping. We present some notations and definitions of the fractional calculus and introduce some basic properties of the SKCPs. Bivariate shifted SKCPs are defined and the operational matrix of fractional integration and some other needed operational matrices are constructed. Our approach uses the properties of bivariate shifted SKCPs to transform the considered problem to a matrix equation without using any collocation points. The main characteristic of this technique is that only a small number of the basic functions is needed to obtain a satisfactory result. An estimation of the error is given in the sense of Sobolev norms. Numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

SU‐D‐210‐03: Limited‐View Multi‐Source Quantitative Photoacoustic Tomography

Purpose:This work is to investigate a novel limited‐view multi‐source acquisition scheme for the direct and simultaneous reconstruction of optical coefficients in quantitative photoacoustic tomography (QPAT), which has potentially improved signal‐to‐noise ratio and reduced data acquisition time.Methods:Conventional QPAT is often considered in two steps: first to reconstruct the initial acoustic pressure from the full‐view ultrasonic data after each optical illumination, and then to quantitatively reconstruct optical coefficients (e.g., absorption and scattering coefficients) from the initial acoustic pressure, using multi‐source or multi‐wavelength scheme.Based on a novel limited‐view multi‐source scheme here, We have to consider the direct reconstruction of optical coefficients from the ultrasonic data, since the initial acoustic pressure can no longer be reconstructed as an intermediate variable due to the incomplete acoustic data in the proposed limited‐view scheme. In this work, based on a coupled photo‐acoustic forward model combining diffusion approximation and wave equation, we develop a limited‐memory Quasi‐Newton method (LBFGS) for image reconstruction that utilizes the adjoint forward problem for fast computation of gradients. Furthermore, the tensor framelet sparsity is utilized to improve the image reconstruction which is solved by Alternative Direction Method of Multipliers (ADMM).Results:The simulation was performed on a modified Shepp‐Logan phantom to validate the feasibility of the proposed limited‐view scheme and its corresponding image reconstruction algorithms.Conclusion:A limited‐view multi‐source QPAT scheme is proposed, i.e., the partial‐view acoustic data acquisition accompanying each optical illumination, and then the simultaneous rotations of both optical sources and ultrasonic detectors for next optical illumination. Moreover, LBFGS and ADMM algorithms are developed for the direct reconstruction of optical coefficients from the acoustic data.Jing Feng and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500).

Variability of flow discharge in lateral inflow-dominated stream channels

Abstract. The influence of the temporal changes in lateral inflow rate on the discharge variability in stream channels is explored through the analysis of the diffusion wave equation (i.e. the linearized Saint-Venant equation). To account for variability and uncertainty, the lateral inflow rate is regarded as a temporal random function. On the basis of the spectral representation theory, analytical expressions for the covariance function and evolutionary power spectral density of the random discharge perturbation process are derived to quantify variability in stream flow discharge induced by the temporal changes in lateral inflow rate. The treatment of the discharge variance (square root of the variance) gives us a quantitative estimate of uncertainty in predictions from the deterministic model. It is found that the discharge variability of stream flow is very large in the downstream reach, indicating large uncertainty anticipated from the use of the deterministic model. A larger temporal correlation scale of inflow rate fluctuations, representing more temporal consistency of fluctuations in inflow rate around the mean, introduces a higher variability in stream flow discharge.

Time-fractional wave-diffusion equation in an inhomogeneous half-space

We consider the fundamental solution of the time-fractional wave-diffusion equation in a three-dimensional half-space medium which contains an inhomogeneity in form of a plane parallel layer. The corresponding Green’s function which is derived by means of the Fourier and Laplace transforms can be accurately and efficiently evaluated without recourse to the Mittag-Leffler or the Fox H-function. Moreover, it is shown that in the one-dimensional case the fundamental solution in an inhomogeneous half-space is no longer a probability density function. In addition, we consider the advection equation for the fractional Laplacian and the Caputo time-fractional derivative of orders on a bounded domain. Simple algorithms for accurate evaluation of the M-Wright function and the Mittag-Leffler function are enclosed at the end of this article.

Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations

In this paper we apply a high order difference scheme and Galerkin spectral technique for the numerical solution of multi-term time fractional partial differential equations. The proposed methods are based on a finite difference scheme in time. The time fractional derivatives which have been described in Caputo’s sense are approximated by a scheme of order O(τ3−α),1<α<2 and the space derivative is discretized with a fourth-order compact finite difference procedure and Galerkin spectral method. We prove the unconditional stability of the compact procedure by coefficient matrix property. The L∞-convergence of the compact finite difference method has been proved by the energy method. Also we obtain an error estimate for Galerkin spectral method. Numerical results are provided to verify the accuracy and efficiency of the proposed schemes.

Discussion of Muskingum method parameter X

The parameter X of the Muskingum method is a physical parameter that reflects the flood peak attenuation and hydrograph shape flattening of a diffusion wave in motion. In this paper, the historic process that hydrologists have undergone to find a physical explanation of this parameter is briefly discussed. Based on the fact that the Muskingum method is the second-order accuracy difference solution to the diffusion wave equation, its numerical stability condition is analyzed, and a conclusion is drawn: X ≤ 0.5 is the uniform condition satisfying the demands for its physical meaning and numerical stability. It is also pointed out that the methods that regard the sum of squares of differences between the calculated and observed discharges or stages as the objective function and the routing coefficients C0, C1 and C2 of the Muskingum method as the optimization parameters cannot guarantee the physical meaning of X.

A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation

The space-time fractional diffusion-wave equation (FDWE) is a generalization of classical diffusion and wave equations which is used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. This paper reports an accurate spectral tau method for solving the two-sided space and time Caputo FDWE with various types of nonhomogeneous boundary conditions. The proposed method is based on shifted Legendre tau (SLT) procedure in conjunction with the shifted Legendre operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided fractional derivatives. We focus primarily on implementing this algorithm in both temporal and spatial discretizations. In addition, convergence analysis is provided theoretically for the Dirichlet boundary conditions, along with graphical analysis for several special cases using other conditions. These suggest that the Legendre Tau method converges exponentially provided that the data in the given FDWE are smooth. Finally, several numerical examples are given to demonstrate the high accuracy of the proposed method.

A compact ADI scheme for the two dimensional time fractional diffusion-wave equation in polar coordinates

In this article, we consider finite difference schemes for two dimensional time fractional diffusion-wave equations on an annular domain. The problem is formulated in polar coordinates and, therefore, has variable coefficients. A compact alternating direction implicit scheme with accuracy order is derived, where τ, h1, h2 are the temporal and spatial step sizes, respectively. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. Numerical experiments are presented to support the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1692–1712, 2015

An ADI Crank–Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave Equation

A new method is formulated and analyzed for the approximate solution of a two-dimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit method based on the Crank---Nicolson method combined with the $$L1$$L1-approximation of the time Caputo derivative of order $$\alpha \in (1,2)$$??(1,2). It is proved that this scheme is stable, and of optimal accuracy in various norms. Numerical experiments demonstrate the predicted global convergence rates and also superconvergence.

Similarity solution to fractional nonlinear space-time diffusion-wave equation

In this article, the so-called fractional nonlinear space-time wave-diffusion equation is presented and discussed. This equation is solved by the similarity method using fractional derivatives in the Caputo, Riesz-Feller, and Riesz senses. Some particular cases are presented and the corresponding solutions are shown by means of 2-D and 3-D plots.