A static memory sparse spectral method for time-fractional PDEs in arbitrary dimensions

We introduce a method which provides accurate numerical solutions to fractional-in-time partial differential equations posed on $[0,T] \times \Omega$ with $\Omega \subset \mathbb{R}^d$ without the excessive memory requirements associated with the nonlocal fractional derivative operator operator. Our approach combines recent advances in the development and utilization of multivariate sparse spectral methods as well as fast methods for the computation of Gauss quadrature nodes with recursive non-classical methods for the Caputo fractional derivative of general fractional order $\alpha > 0$. An attractive feature of the method is that it has minimal theoretical overhead when using it on any domain $\Omega$ on which an orthogonal polynomial basis is already available. We discuss the memory requirements of the method, present several numerical experiments demonstrating the method's performance in solving time-fractional PDEs on intervals, triangles and disks and derive error bounds which suggest sensible convergence strategies. As an important model problem for this approach we consider a type of wave equation with time-fractional dampening related to acoustic waves in viscoelastic media with applications in the physics of medical ultrasound and outline future research steps required to use such methods for the reverse problem of image reconstruction from sensor data.Comment: 28 pages, 13 figure

Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.Comment: submitted to Siam SIA

Wave propagation in a fractional viscoelastic Andrade medium: diffusive approximation and numerical modeling

This study focuses on the numerical modeling of wave propagation in fractionally-dissipative media. These viscoelastic models are such that the attenuation is frequency dependent and follows a power law with non-integer exponent. As a prototypical example, the Andrade model is chosen for its simplicity and its satisfactory fits of experimental flow laws in rocks and metals. The corresponding constitutive equation features a fractional derivative in time, a non-local term that can be expressed as a convolution product which direct implementation bears substantial memory cost. To circumvent this limitation, a diffusive representation approach is deployed, replacing the convolution product by an integral of a function satisfying a local time-domain ordinary differential equation. An associated quadrature formula yields a local-in-time system of partial differential equations, which is then proven to be well-posed. The properties of the resulting model are also compared to those of the original Andrade model. The quadrature scheme associated with the diffusive approximation, and constructed either from a classical polynomial approach or from a constrained optimization method, is investigated to finally highlight the benefits of using the latter approach. Wave propagation simulations in homogeneous domains are performed within a split formulation framework that yields an optimal stability condition and which features a joint fourth-order time-marching scheme coupled with an exact integration step. A set of numerical experiments is presented to assess the efficiency of the diffusive approximation method for such wave propagation problems.Comment: submitted to Wave Motio

The sine and cosine diffusive representations for the Caputo fractional derivative

As we are aware, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order $0<\alpha<1$. Error analysis of the newly presented methods together with some numerical examples are provided at the end

Good (and Not So Good) practices in computational methods for fractional calculus

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results

Trends, directions for further research, and some open problems of fractional calculus

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.info:eu-repo/semantics/publishedVersio

Fractional equations and diffusive systems: an overview

The aim of this discussion is to give a broad view of the links between fractional differential equations (FDEs) or fractional partial differential equations (FPDEs) and so-called diffusive representations (DR). Many aspects will be investigated: theory and numerics, continuous time and discrete time, linear and nonlinear equations, causal and anti-causal operators, optimal diffusive representations, fractional Laplacian. Many applications will be given, in acoustics, continuum mechanics, electromagnetism, identification, ..

Efficient high-order methods for solving fractional differential equations of order α ∈ (0, 1) using fast convolution and applications in wave propagation

In this work we develop a means to rapidly and accurately compute the Caputo fractional derivative of a function, using fast convolution. The key element to this approach is the compression of the fractional kernel into a sum of M decaying exponentials, where M is minimal. Specifically, after N time steps we find M= O (log N) leading to a scheme with O (N log N) complexity. We illustrate our method by solving the fractional differential equation representing the Kelvin-Voigt model of viscoelasticity, and the partial differential equations that model the propagation of electromagnetic pulses in the Cole-Cole model of induced dielectric polarization

Compressing the memory variables in constant-Q viscoelastic wave propagation via an improved sum-of-exponentials approximation

Earth introduces strong attenuation and dispersion to propagating waves. The time-fractional wave equation with very small fractional exponent, based on Kjartansson's constant-Q theory, is widely recognized in the field of geophysics as a reliable model for frequency-independent Q anelastic behavior. Nonetheless, the numerical resolution of this equation poses considerable challenges due to the requirement of storing a complete time history of wavefields. To address this computational challenge, we present a novel approach: a nearly optimal sum-of-exponentials (SOE) approximation to the Caputo fractional derivative with very small fractional exponent, utilizing the machinery of generalized Gaussian quadrature. This method minimizes the number of memory variables needed to approximate the power attenuation law within a specified error tolerance. We establish a mathematical equivalence between this SOE approximation and the continuous fractional stress-strain relationship, relating it to the generalized Maxwell body model. Furthermore, we prove an improved SOE approximation error bound to thoroughly assess the ability of rheological models to replicate the power attenuation law. Numerical simulations on constant-Q viscoacoustic equation in 3D homogeneous media and variable-order P- and S- viscoelastic wave equations in 3D inhomogeneous media are performed. These simulations demonstrate that our proposed technique accurately captures changes in amplitude and phase resulting from material anelasticity. This advancement provides a significant step towards the practical usage of the time-fractional wave equation in seismic inversion.Comment: 28 pages, 52 figures, 6 table