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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. • Lossless model reduction for the seismic wave equation in earthquake modeling. • Fast and accurate algorithm for solving 3D P- and S- viscoelastic wave equation. • Improved theoretical analysis of the SOE approximation for the fractional derivative. • Nearly optimal construction of the SOE approximation using GGQ. [ABSTRACT FROM AUTHOR]

Highlights • A new finite difference scheme for space fractional equations is constructed. • The scheme is unconditionally stable for time dependent problems. • The scheme is fourth order accurate in space. • The same idea could be used to construct even higher order difference methods. Abstract A weighted and shifted difference formula is constructed based on the Lubich operators, which gives a forth-order and unconditionally stable difference scheme for the Cauchy problem of space fractional diffusion equations. The novelty of the proposed method here is that only four weighted parameters are required, compared to eight parameters used in the previous work, to achieve the fourth-order accuracy and to ensure the stability at the same time. To verify the efficiency of the proposed scheme, several numerical experiments for both one-dimensional and two-dimensional fractional diffusion problems are provided. [ABSTRACT FROM AUTHOR]

• A short-memory operator splitting scheme for constant-Q wave equation is proposed. • The short memory principle is based on the scaling in the Laguerre spectral method. • A solid theoretical analysis of the Yuan-Agrawal method is provided. • An efficient algorithm for calculating the Mainardi function is given. We propose a short-memory operator splitting scheme for solving the constant-Q wave equation, where the fractional stress-strain relation contains multiple Caputo fractional derivatives with order much smaller than 1. The key is to exploit its extension problem by converting the flat singular kernels into strongly localized ones, so that the major contribution of weakly singular integrals over a semi-infinite interval can be captured by a few Laguerre functions with proper asymptotic behavior. Despite its success in reducing both memory requirement and arithmetic complexity, we show that numerical accuracy under prescribed memory variables may deteriorate in time due to the dynamical increments of projection errors. Fortunately, it can be considerably alleviated by introducing a suitable scaling factor β > 1 and pushing the collocation points closer to origin. An operator splitting scheme is introduced to solve the resulting set of equations, where the auxiliary dynamics can be solved exactly, so that it gets rid of the numerical stiffness and discretization errors. Numerical experiments on both 1-D diffusive wave equation and 2-D constant-Q P - and S -wave equations are presented to validate the accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

• Consider a multidimensional time-dependent space-fractional equation to model solute transport in heterogeneous porous media. • The equation involves all directions in its fractional derivative rather than only coordinate directions. • We derive the fast control-volume schemes, these schemes naturally have second order accuracy in space. • A fast Krylov subspace iterative solver for control volume schemes is proposed. We develop control volume methods for two-dimensional time-dependent advection-dominated directional space-fractional advection-dispersion equations with the directional space-fractional derivative weighted in all the directions by a probability measure in the unit circle, which are used to model the anisotropic superdiffusive transport of solutes in groundwater moving through subsurface heterogeneous porous media. We develop a fast upwind control volume method for the governing equation to eliminate the spurious numerical oscillations that often occur in space-centered numerical discretizations of advection term, which are relatively straightforward to implement. We also develop a Eulerian-Lagrangian control-volume method for the governing equation, which symmetrizes the governing equation by combining the time-derivative term and the advection term into a material derivative term along characteristic curves. Both methods are locally mass-conservative, which are essential in these applications. Due to the nonlocal nature of the directional space-fractional differential operators, corresponding numerical discretizations usually generate full stiffness matrices. Conventional direct solvers tend to require O (N 2) memory requirement and have O (N 3) computational complexity per time step, where N is the number of spatial unknowns, which is computationally significantly more expensive than the numerical approximations of integer-order advection-diffusion equations. Based on the analysis of the structure of stiffness matrix, we propose a fast Krylov subspace iterative solver to accelerate the numerical approximations of both the upwind and Eulerian-Lagrangian control volume methods, which reduce computational complexity from O (N 3) by a direct solver to O (N log ⁡ N) per Krylov subspace iteration per time step and a memory requirement from O (N 2) to O (N). Numerical results are presented to show the utility of the methods. [ABSTRACT FROM AUTHOR]