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In this study, we revisit the spectral transfer model for the turbulent intensity in passive scalar transport (under large-scale anisotropic forcing), and a subsequent modification to the scaling of scalar variance cascade is presented. From the modified spectral transfer model, we obtain a revised scalar transport model using a fractional-order Laplacian operator that facilitates the robust inclusion of the non-local effects originating from large-scale anisotropy transferred across the multitude of scales in the turbulent cascade. We provide an a priori estimate for the non-local model based on the scaling analysis of the scalar spectrum, and later examine our developed model through direct numerical simulation. We present a detailed analysis on the evolution of the scalar variance, high-order statistics of the scalar gradient and important two-point statistical metrics of the turbulent transport to make a comprehensive comparison between the non-local model and its standard version. Finally, we present an analysis that seamlessly reconciles the similarities between the developed model with the fractional-order subgrid-scale scalar flux model for large-eddy simulation (Akhavan-Safaei et al., J. Comput. Phys., vol. 446, 2021, 110571) when the filter scale approaches the dissipative scales of turbulent transport. In order to perform this task, we employ a Gaussian process regression model to predict the model coefficient for the fractional-order subgrid model.

Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order α ∈ ( 0 , 1 ] and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in the literature. Furthermore, the size of the stability regions increase with the decrease of fractional order values, which is suitable for stiff problems. We also rewrite the resulting IMEX methods in the language of nonlinear Toeplitz systems, where we employ a fast inversion scheme to achieve a computational complexity of O ( N log ⁡ N ) , where N denotes the number of time-steps. Our computational results demonstrate that the developed schemes can achieve global first- and second-order accuracy for highly-oscillatory stiff/nonlinear problems with singularities.

We develop a two-stage computational framework for robust and accurate time-integration of multi-term linear/nonlinear fractional differential equations. In the first stage, we formulate a self-sin...

Extensive experimental evidence highlight that scalar turbulence exhibits anomalous diffusion and stronger intermittency levels at small scales compared to that in fluid turbulence. This renders the corresponding subgrid-scale dynamics modeling for scalar turbulence a greater challenge to date. We develop a new large eddy simulation (LES) paradigm for efficiently and dynamically nonlocal LES modeling of the scalar turbulence. To this end, we formulate the underlying nonlocal model starting from the filtered Boltzmann kinetic transport equation, where the divergence of subgrid-scale scalar fluxes emerges as a fractional-order Laplacian term in the filtered advection-diffusion model, coding the corresponding supper-diffusive nature of scalar turbulence. Subsequently, we develop a robust data-driven algorithm for estimation of the fractional (non-integer) Laplacian exponent, where we on-the-fly calculate the corresponding model coefficient employing a new dynamic procedure. Our $\textit{a priori}$ tests show that our new dynamically nonlocal LES paradigm provides better agreements with the ground-truth filtered DNS data in comparison to the conventional static and dynamic Prandtl-Smagorisnky models. Moreover, in order to analyze the numerical stability and assessing the model's performance, we carry out a comprehensive $\textit{a posteriori}$ tests. They unanimously illustrate that our new model considerably outperforms other existing functional models, correctly predicting the backscattering phenomena at the same time and providing higher correlations at small-to-large filter sizes. We conclude that our proposed nonlocal subgrid-scale model for scalar turbulence is amenable for coarse LES and VLES frameworks even with strong anisotropies, applicable to environmental applications., Comment: 17 pages, 5 figures

The presence of nonlocal interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modeling framework for developing nonlocal LES subgrid-scale models, starting from the kinetic transport. We employ a tempered L\'evy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\Delta+\lambda)^{\alpha} (\cdot)$, for $\alpha \in (0,1)$, $\alpha \neq \frac{1}{2}$, and $\lambda>0$ in the filtered Navier-Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau 1994). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model., Comment: 28 pages, 10 figures, 3 tables

Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin–Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance.

The vibration suppression of fractionally damped thin rectangular plates is studied. The plate has simply supported edges and is subjected to a concentrated harmonic loading. The vibration suppress...

Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model coefficients. We formulate a sensitivity analysis of fractional models by developing a fractional sensitivity equation method. We obtain the adjoint fractional sensitivity equations, in which we present a fractional operator associated with logarithmic-power law kernel. We further construct a gradient-based optimization algorithm to compute an accurate parameter estimation in fractional model construction. We develop a fast, stable, and convergent Petrov–Galerkin spectral method to numerically solve the coupled system of original fractional model and its corresponding adjoint fractional sensitivity equations.

From their birth in the manufacturing process, materials inherently contain defects that affect the mechanical behavior across multiple length and time-scales, including vacancies, dislocations, voids and cracks. Understanding, modeling, and real-time simulation of the underlying stochastic micro-structure defect evolution is therefore vital towards multi-scale coupling and propagating numerous sources of uncertainty from atomistic to eventually aging continuum mechanics. We develop a graph-based surrogate model of dislocation glide for computation of stochastic dislocation mobility. We model an edge dislocation as a random walker, jumping between neighboring nodes of a graph following a Poisson stochastic process. The network representation functions as a coarse-graining of a molecular dynamics simulation that provides dislocation trajectories for an empirical computation of jump rates. With this construction, we recover the original atomistic mobility estimates, with remarkable computational speed-up and accuracy. Furthermore, the underlying stochastic process provides the statistics of dislocation mobility associated to the original molecular dynamics simulation, allowing an efficient propagation of material parameters and uncertainties across the scales.

We developed a novel autonomously dynamic nonlocal turbulence model for the large and very large eddy simulation (LES, VLES) of the homogeneous isotropic turbulent flows (HIT). The model is based on a generalized (integer-to-noninteger) order Laplacian of the filtered velocity field, and a novel dynamic model has been formulated to avoid the need for tuning the model constant. Three data-driven approaches were introduced for the determination of the fractional-order to have a model which is totally free of any tuning parameter. Our analysis includes both the $\textit{a priori}$ and the $\textit{a posteriori}$ tests. In the former test, using a high-fidelity and well-resolved dataset from direct numerical simulations (DNS), we computed the correlation coefficients for the stress components of the subgrid-scale (SGS) stress tensor and the one we get directly from the DNS results. Moreover, we compared the probability density function of the ensemble-averaged SGS forces for different filter sizes. In the latter, we employed our new model along with other conventional models including static and dynamic Smagorinsky into our pseudo-spectral solver and tested the final predicted quantities. The results of the newly developed model exhibit an expressive agreement with the ground-truth DNS results in all components of the SGS stress and forces. Also, the model exhibits promising results in the VLES region as well as the LES region, which could be remarkably important for the cost-efficient nonlocal turbulence modelings e.g., in meteorological and environmental applications., 26 pages, 10 figures

We study the flow dynamics inside a high-speed rotating cylinder after introducing strong symmetry-breaking disturbance factors at cylinder wall motion. We propose and formulate a mathematically robust stochastic model for the rotational motion of cylinder wall alongside the stochastic representation of incompressible Navier-Stokes equations. We employ a comprehensive stochastic computational fluid dynamics framework combining spectral/$hp$ element method and probabilistic collocation method to obtain high-fidelity realizations of our mathematical model in order to quantify the propagation of parametric uncertainty for dynamics-representative quantities of interests. We observe that the modeled symmetry-breaking disturbances cause a flow instability arising from the wall. Utilizing global sensitivity analysis approaches, we identify the dominant source of uncertainty in our proposed model. We next perform a qualitative and quantitative statistical analysis on the fluctuating fields characterizing the fingerprints and measures of intense and rapidly evolving non-Gaussian behavior through space and time. We claim that such non-Gaussian statistics essentially emerge and evolve due to an intensified presence of coherent vortical motions initially triggered by the flow instability due to symmetry-breaking rotation of the cylinder. We show that this mechanism causes memory effects in the flow dynamics in a way that noticeable anomaly in the time-scaling of enstrophy record is observed in the long run apart from the onset of instability. Our findings suggest an effective strategy to exploit controlled flow instabilities in order to enhance the turbulent mixing in engineering applications., 40 pages, 13 figures

Failure in brittle materials led by the evolution of micro- to macro-cracks under repetitive or increasing loads is often catastrophic with no significant plasticity to advert the onset of fracture. Early failure detection with respective location are utterly important features in any practical application, both of which can be effectively addressed using artificial intelligence. In this paper, we develop a supervised machine learning (ML) framework to predict failure in an isothermal, linear elastic and isotropic phase-field model for damage and fatigue of brittle materials. Time-series data of the phase-field model is extracted from virtual sensing nodes at different locations of the geometry. A pattern recognition scheme is introduced to represent time-series data/sensor nodes responses as a pattern with a corresponding label, integrated with ML algorithms, used for damage classification with identified patterns. We perform an uncertainty analysis by superposing random noise to the time-series data to assess the robustness of the framework with noise-polluted data. Results indicate that the proposed framework is capable of predicting failure with acceptable accuracy even in the presence of high noise levels. The findings demonstrate satisfactory performance of the supervised ML framework, and the applicability of artificial intelligence and ML to a practical engineering problem, i.,e, data-driven failure prediction in brittle materials.

Filtering the passive scalar transport equation in the large-eddy simulation (LES) of turbulent transport gives rise to the closure term corresponding to the unresolved scalar flux. Understanding and respecting the statistical features of subgrid-scale (SGS) flux is a crucial point in robustness and predictability of the LES. In this work, we investigate the intrinsic nonlocal behavior of the SGS passive scalar flux through studying its two-point statistics obtained from the filtered direct numerical simulation (DNS) data for passive scalar transport in homogeneous isotropic turbulence (HIT). Presence of long-range correlations in true SGS scalar flux urges to go beyond the conventional local closure modeling approaches that fail to predict the non-Gaussian statistical features of turbulent transport in passive scalars. Here, we propose an appropriate statistical model for microscopic SGS motions by taking into account the filtered Boltzmann transport equation (FBTE) for passive scalar. In FBTE, we approximate the filtered equilibrium distribution with an $\alpha$-stable Levy distribution that essentially incorporates a power-law behavior to resemble the observed nonlocal statistics of SGS scalar flux. Generic ensemble-averaging of such FBTE lets us formulate a continuum level closure model for the SGS scalar flux appearing in terms of fractional-order Laplacian that is inherently nonlocal. Through a data-driven approach, we infer the optimal version of our SGS model using the high-fidelity data for the two-point correlation function between the SGS scalar flux and filtered scalar gradient, and sparse linear regression. In an \textit{a priori} test, the optimal fractional-order model yields a promising performance in reproducing the probability distribution function (PDF) of the SGS dissipation of the filtered scalar variance compared to its true PDF obtained from the filtered DNS data., Comment: 20 pages, 6 figures

Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov–Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure.

We develop a pseudo-spectral method of Petrov–Galerkin sense, employing nodal expansions in the weak formulation of distributed-order fractional partial differential equations. We define the underl...

We develop a new $C^{\,0}$-continuous Petrov-Galerkin spectral element method for one-dimensional fractional elliptic problems of the form ${}_{0}{\mathcal{D}}_{x}^{\alpha} u(x) - \lambda u(x) = f(x)$, $\alpha \in (1,2]$, subject to homogeneous boundary conditions. We employ the standard (modal) spectral element bases and the Jacobi poly-fractonomials as the test functions [1]. We formulate a new procedure for assembling the global linear system from elemental (local) mass and stiffness matrices. The Petrov-Galerkin formulation requires performing elemental (local) construction of mass and stiffness matrices in the standard domain only once. Moreover, we efficiently obtain the non-local (history) stiffness matrices, in which the non-locality is presented analytically for uniform grids. We also investigate two distinct choices of basis/test functions: i) local basis/test functions, and ii) local basis with global test functions. We show that the former choice leads to a better-conditioned system and accuracy. We consider smooth and singular solutions, where the singularity can occur at boundary points as well as in the interior domain. We also construct two non-uniform grids over the whole computational domain in order to capture singular solutions. Finally, we perform a systematic numerical study of non-local effects via full and partial history fading in order to further enhance the efficiency of the scheme., Comment: 40 pages, 15 figures, 7 Tables

Fractional calculus has been used to model various hydrologic processes for 15 years. Yet, there are still major gaps between real-world hydrologic dynamics and fractional-order partial differential equations (fPDEs). In addition, the applicability of fPDEs in the broad field of Earth dynamics remains obscure. This study first reviews previous applications and then identifies new research directions for fPDEs simulating non-Fickian transport in both surface and subsurface hydrology. We then explore the applicability of fractional calculus in various anomalous dynamics with a wide range of spatiotemporal scales observed in the solid Earth, including internal dynamics (such as inner core rotation, outer core flow, mantle convection, and crustal deformation), large-scale surface dynamics (in fluvial, Aeolian, and glacial systems), and small vertical-scale surface kinetics (in crystal growth, rock/mineral weathering, and pedogenesis), where driven forces, previous modeling approaches, and the details of anomalous dynamics are analyzed. Results show that the solid Earth can provide an ideal and diverse base for the application of fractional calculus and fPDEs. Complex dynamics within and across spatiotemporal scales, multi-scale intrinsic heterogeneity, and intertwined controlling factors for dynamic processes in the solid Earth can motivate the application of fPDEs. Challenges for the future application of fPDEs in Earth systems are also discussed, including poor parameter predictability, the lack of mathematical specification of bounded fractional diffusion, lack of intermediate-scale geologic information in parsimonious and upscaling models, and a lack of models for multi-phase and coupled processes. Substantial extension of fPDE models is needed for the development of next-generation, solid Earth dynamic models, where potential solutions are discussed based on our experience gained in the development and application of fractional calculus and fPDEs over the last decade. Therefore, the current bottleneck in the application of fractional calculus in hydrologic sciences should not be the end of a promising stochastic approach, but could be the early stage of a decade-long effort filled with multiple new research and application directions in geology. This conclusion may shed light on the bottleneck challenging stochastic hydrogeology, where the advanced stochastic models (with more than 3500 journal publications in the last three decades) have not significantly impacted the practice of groundwater flow and transport modeling.

Distributed order fractional operators offer a rigorous tool for mathematical modelling of multi-physics phenomena, where the differential orders are distributed over a range of values rather than being just a fixed integer/fraction as it is in standard/fractional ODEs/PDEs. We develop two spectrally-accurate schemes, namely a Petrov-Galerkin spectral method and a spectral collocation method for distributed order fractional differential equations. These schemes are developed based on the fractional Sturm-Liouville eigen-problems (FSLPs). In the Petrov-Galerkin method, we employ fractional (non-polynomial) basis functions, called \textit{Jacobi poly-fractonomials}, which are the eigenfunctions of the FSLP of first kind, while, we employ another space of test functions as the span of poly-fractonomial eigenfunctions of the FSLP of second kind. We define the underlying \textit{distributed Sobolev space} and the associated norms, where we carry out the corresponding discrete stability and error analyses of the proposed scheme. In the collocation scheme, we employ fractional (non-polynomial) Lagrange interpolants satisfying the Kronecker delta property at the collocation points. Subsequently, we obtain the corresponding distributed differentiation matrices to be employed in the discretization of the strong problem. We perform systematic numerical tests to demonstrate the efficiency and conditioning of each method.

We present a new tunably accurate Laguerre Petrov--Galerkin spectral method for solving linear multiterm fractional initial value problems with derivative orders at most one and constant coefficients on the half line. Our method results in a matrix equation of special structure which can be solved in $\mathcal{O}(N \log N)$ operations. We also take advantage of recurrence relations for the generalized associated Laguerre functions (GALFs) in order to derive explicit expressions for the entries of the stiffness and mass matrices, which can be factored into the product of a diagonal matrix and a lower-triangular Toeplitz matrix. The resulting spectral method is efficient for solving multiterm fractional differential equations with arbitrarily many terms, which we demonstrate by solving a fifty-term example. We apply this method to a distributed order differential equation, which is approximated by linear multiterm equations through the Gauss--Legendre quadrature rule. We provide numerical examples demonstra...

Coherent structures/motions in turbulence inherently give rise to intermittent signals with sharp peaks, heavy-skirt, and skewed distributions of velocity increments, highlighting the non-Gaussian nature of turbulence. That suggests that the spatial nonlocal interactions cannot be ruled out of the turbulence physics. Furthermore, filtering the Navier-Stokes equations in the large eddy simulation of turbulent flows would further enhance the existing nonlocality, emerging in the corresponding subgrid scale fluid motions. That urges the development of new nonlocal closure models, which respect the corresponding non-Gaussian statistics of the subgrid stochastic motions. To this end and starting from the filtered Boltzmann equation, we model the corresponding equilibrium distribution function with a \textit{L\'evy}-stable distribution, leading to the proposed fractional-order modeling of subgrid-scale stresses. We approximate the filtered equilibrium distribution function with a power-law term, and derive the corresponding filtered Navier-Stokes equations. Subsequently in our functional modeling, the divergence of subgrid-scale stresses emerges as a single-parameter fractional Laplacian, $(-\Delta)^{\alpha}(\cdot)$, $\alpha \in (0,1]$, of the filtered velocity field. The only model parameter, i.e., the fractional exponent, appears to be strictly depending on the filter-width and the flow Reynolds number. We furthermore explore the main physical and mathematical properties of the proposed model under a set of mild conditions. Finally, the introduced model undergoes \textit{a priori} evaluations based on the direct numerical simulation database of forced and decaying homogeneous isotropic turbulent flows at relatively high and moderate Reynolds numbers, respectively. Such analysis provides a comparative study of predictability and performance of the proposed fractional model.

We develop a thermodynamically consistent, fractional visco-elasto-plastic model coupled with damage for anomalous materials. The model utilizes Scott-Blair rheological elements for both visco- elastic/plastic parts. The constitutive equations are obtained through Helmholtz free-energy potentials for Scott-Blair elements, together with a memory-dependent fractional yield function and dissipation inequalities. A memory-dependent Lemaitre-type damage is introduced through fractional damage energy release rates. For time-fractional integration of the resulting nonlinear system of equations, we develop a first-order semi-implicit fractional return-mapping algorithm. We also develop a finite-difference discretization for the fractional damage energy release rate, which results into Hankel-type matrix–vector operations for each time-step, allowing us to reduce the computational complexity from O ( N 3 ) to O ( N 2 ) through the use of Fast Fourier Transforms. Our numerical results demonstrate that the fractional orders for visco-elasto-plasticity play a crucial role in damage evolution, due to the competition between the anomalous plastic slip and bulk damage energy release rates.

We first formulate a fractional class of explicit Adams-Bashforth (A-B) and implicit Adams-Moulton (A-M) methods of first- and second-order accuracy for the time-integration of D t ? 0 C u ( x , t ) = g ( t ; u ) , ? ? ( 0 , 1 , where D t ? 0 C denotes the fractional derivative in the Caputo sense. In this fractional setting and in contrast to the standard Adams methods, an extra history load term emerges and the associated weight coefficients are ?-dependent. However when ? = 1 , the developed schemes reduce to the well-known A-B and A-M methods with standard coefficients. Hence, in terms of scientific computing, our approach constitutes a minimal modification of the existing Adams libraries. Next, we develop an implicit-explicit (IMEX) splitting scheme for linear and nonlinear fractional PDEs of a general advection-reaction-diffusion type, and we apply our scheme to the time-space fractional Keller-Segel chemotaxis system. In this context, we evaluate the nonlinear advection term explicitly, employing the fractional A-B method in the prediction step, and we treat the corresponding diffusion term implicitly in the correction step using the fractional A-M scheme. Moreover, we perform the corresponding spatial discretization by employing an efficient and spectrally-accurate fractional spectral collocation method. Our numerical experiments exhibit the efficiency of the proposed IMEX scheme in solving nonlinear fractional PDEs.

While several high-order methods have been developed for fractional PDEs (FPDEs) with fixed order, there are no such methods for FPDEs with field-variable order. These equations allow multiphysics simulations seamlessly, e.g. from diffusion to sub-diffusion or from wave dynamics transitioning to diffusion, by simply varying the fractional order as a function of space or time. We develop an exponentially accurate fractional spectral collocation method for solving linear/nonlinear FPDEs with field-variable order. Following the spectral theory, developed in 1] for fractional Sturm-Liouville eigenproblems, we introduce a new family of interpolants, called left-/right-sided and central fractional Lagrange interpolants. We employ the fractional derivatives of (left-/right-sided) Riemann-Liouville and Riesz type and obtain the corresponding fractional differentiation matrices by collocating the field-variable fractional orders. We solve several FPDEs including time- and space-fractional advection-equation, time- and space-fractional advection-diffusion equation, and finally the space-fractional Burgers' equation to demonstrate the performance of the method. In addition, we develop a spectral penalty method for enforcing inhomogeneous initial conditions. Our numerical results confirm the exponential-like convergence of the proposed fractional collocation methods.

Existing numerical methods for fractional PDEs suffer from low accuracy and inefficiency in dealing with three-dimensional problems or with long-time integrations. We develop a unified and spectrally accurate Petrov–Galerkin (PG) spectral method for a weak formulation of the general linear Fractional Partial Differential Equations (FPDEs) of the form 0Dt2τu+∑j=1dcj[ajDxj2μju]+γu=f, where 2τ2τ, μj∈(0,1)μj∈(0,1), in a (1+d1+d)-dimensional space–time domain subject to Dirichlet initial and boundary conditions. We perform the stability analysis (in 1-D) and the corresponding convergence study of the scheme (in multi-D). The unified PG spectral method applies to the entire family of linear hyperbolichyperbolic-, parabolicparabolic- and ellipticelliptic-like equations. We develop the PG method based on a new spectral theory for fractional Sturm–Liouville problems (FSLPs), recently introduced in Zayernouri and Karniadakis (2013). Specifically, we employ the eigenfunctions of the FSLP of first kind (FSLP-I), called Jacobi poly-fractonomials, as temporal/spatial bases. Next, we construct a different space for test functions from poly-fractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Besides the high-order spatial accuracy of the PG method, we demonstrate its efficiency and spectral accuracy in time-integration schemes for solving time-dependent FPDEs as well, rather than employing algebraically accurate traditional methods, especially when 2τ=12τ=1. Finally, we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices, which reduces the computational cost drastically. We demonstrate that this framework can reduce to hyperbolic FPDEs such as time- and space-fractional advection (TSFA), parabolic FPDEs such as time- and space-fractional diffusion (TSFD) model, and elliptic FPDEs such as fractional Helmholtz/Poisson equations with the same ease and cost. Several numerical tests confirm the efficiency and spectral convergence of the unified PG spectral method for the aforementioned families of FPDEs. Moreover, we demonstrate the computational efficiency of the new approach in higher-dimensions e.g., (1+3), (1+5) and (1+9)-dimensional problems.

Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Levy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable Levy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm--Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order $\nu \in (0,2)$. In the regular case, the corresponding tempered differential operators are associated with tempering functions $p_I(x) = \exp(2\tau) $ and $p_{II}(x) = \exp(-2\tau)$, $\tau \geq 0$, respectively, in the regular TFSLP-I...

We present the stability and error analysis of the unified Petrov–Galerkin spectral method, developed in [1] , for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any ( 1 + d )-dimensional space-time hypercube, d = 1 , 2 , 3 , ⋯ , subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method. Finally, we perform several numerical simulations to compare the theoretical and computational rates of convergence.

Temporally-distributed fractional partial differential equations appear as rigorous mathematical models that solve the probability density function of non-Markovian processes coding multi-physics diffusion-to-wave and multi-rate ultra slow-to-super diffusion dynamics (Chechkin et al, Phys Rev E 66(4):046129, 2002). We develop a Petrov-Galerkin spectral method for high dimensional temporally-distributed fractional partial differential equations with two-sided derivatives in a space-time hypercube. We employ Jacobi poly-fractonomials given in (Zayernouri and Karniadakis, J Comput Phys 252:495–517, 2013) and Legendre polynomials as the temporal and spatial basis/test functions, respectively. Moreover, we formulate a fast linear solver for the corresponding Lyapunov system. Furthermore, we perform the corresponding discrete stability and error analysis of the numerical scheme. Finally, we carry out several numerical test cases to examine the efficiency and accuracy of the method.

We develop a unified Petrov–Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form D t 2 τ 0 u + ∑ i = 1 d [ c l i a i D x i 2 μ i u + c r i x i D b i 2 μ i u ] + γ u = ∑ j = 1 d [ κ l j a j D x j 2 ν j u + κ r j x j D b j 2 ν j u ] + f , where 2 τ ∈ ( 0 , 2 ) , 2 τ ≠ 1 , 2 μ i ∈ ( 0 , 1 ) and 2 ν j ∈ ( 1 , 2 ) , in a ( 1 + d )-dimensional space–time hypercube, d = 1 , 2 , 3 , ⋯ , subject to homogeneous Dirichlet initial/boundary conditions. We employ the eigenfunctions of the fractional Sturm–Liouville eigen-problems of the first kind in [1] , called Jacobi poly-fractonomials, as temporal bases, and the eigen-functions of the boundary-value problem of the second kind as temporal test functions. Next, we construct our spatial basis/test functions using Legendre polynomials, yielding mass matrices being independent of the spatial fractional orders ( μ i , ν j , i , j = 1 , 2 , ⋯ , d ). Furthermore, we formulate a novel unified fast linear solver for the resulting high-dimensional linear system based on the solution of generalized eigen-problem of spatial mass matrices with respect to the corresponding stiffness matrices, hence, making the complexity of the problem optimal, i.e., O ( N d + 2 ) . We carry out several numerical test cases to examine the CPU time and convergence rate of the method. The corresponding stability and error analysis of the Petrov–Galerkin method are carried out in [2] .

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 47 (2013), pp. 2108--2131] for fractional Sturm--Liouville eigenproblems. Subsequently, we obtain the corresponding fractional differentiation matrices, and we solve a number of linear FODEs in addition to linear and nonlinear FPDEs to investigate the numerical performance of the fractional collocation method. We first examine space-fractional advection-diffusion problem and generalized space-fractional multiterm FODEs. Next, we solve FPDEs, including the time- and space-fractional advection-diffusion equation, time- and space-fractional multiterm FPDEs, and finally the space...

We first develop a spectrally accurate Petrov--Galerkin spectral method for fractional delay differential equations (FDDEs). This scheme is developed based on a new spectral theory for fractional Sturm--Liouville problems (FSLPs), which has been recently presented in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495--517]. Specifically, we obtain solutions to FDDEs in terms of new nonpolynomial basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of the first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of the second kind (FSLP-II). We prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the PG spectral method. In contrast to standard (nondelay) fractional differential equations, the delay character of FDDEs might induce solutions, which are either nonsmooth or piecewise smooth. In order to effectively trea...

We develop spectral element methods for a time- and space-fractional advection equation of the form $\prescript{}{0}{\mathcal{D}}^{\tau}_{t} u(x,t) + \theta\,\, \prescript{}{0}{\mathcal{D}}^{\nu}_{x} u(x,t) = f(x,t)$, of order $\tau\in (0,1]$, $\nu \in (0,1)$, subject to Dirichlet initial/boundary conditions. We present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms, instead of employing traditional low-order time-integration methods. To this end, we first develop a Petrov--Galerkin in time and discontinuous Galerkin in space (PG-DG) method, where we carry out the time-integration using a single time-domain spectral method (SM), and we perform the spatial discretization using the discontinuous spectral/$hp$ element method (DSEM). This scheme also leads to a more efficient time-integration when the time-derivative is integer-order, i.e., $\tau=1$. We develop the SM-DSEM scheme based on a new spectral theory for fractional Sturm--Liouville problems...

Current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy. Here, we present different methods for fractional ODEs that lead to exponentially fast decay of the error. First, we develop a Petrov-Galerkin (PG) spectral method for Fractional Initial-Value Problems (FIVPs) of the form D t ? 0 u ( t ) = f ( t ) and Fractional Final-Value Problems (FFVPs) D T ? t u ( t ) = g ( t ) , where ? ? ( 0 , 1 ) , subject to Dirichlet initial/final conditions. These schemes are developed based on a new spectral theory for fractional Sturm-Liouville problems (FSLPs), which has been recently developed in 1]. Specifically, we obtain solutions to FIVPs and FFVPs in terms of the new fractional (non-polynomial) basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Subsequently, we develop a Discontinuous Spectral Method (DSM) of Petrov-Galerkin sense for the aforementioned FIVPs and FFVPs, where the basis functions do not satisfy the initial/final conditions. Finally, we extend the DSM scheme to a Discontinuous Spectral Element Method (DSEM) for efficient longer time-integration and adaptive refinement. In these discontinuous schemes, we employ the asymptotic eigensolutions to FSLP-I & -II, which are of Jacobi polynomial forms, as basis and test functions. Our numerical tests confirm the exponential/algebraic convergence, respectively, in p- and h-refinements, for various test cases with integer- and fractional-order solutions.

We first consider a regular fractional Sturm-Liouville problem of two kinds RFSLP-I and RFSLP-II of order ? ? ( 0 , 2 ) . The corresponding fractional differential operators in these problems are both of Riemann-Liouville and Caputo type, of the same fractional order µ = ? / 2 ? ( 0 , 1 ) . We obtain the analytical eigensolutions to RFSLP-I & -II as non-polynomial functions, which we define as Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with RFSLP-I & -II. Subsequently, we extend the fractional operators to a new family of singular fractional Sturm-Liouville problems of two kinds, SFSLP-I and SFSLP-II. We show that the primary regular boundary-value problems RFSLP-I & -II are indeed asymptotic cases for the singular counterparts SFSLP-I & -II. Furthermore, we prove that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. In addition, we obtain the eigen-solutions to SFSLP-I & -II analytically, also as non-polynomial functions, hence completing the whole family of the Jacobi poly-fractonomials. In numerical examples, we employ the new poly-fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results.

We present an efficient computational method to model fluid flow in the presence of random wall roughness. A random flow domain is represented by a stochastic indicator function having a smoothed profile perpendicular to roughness, and the random domain is discretized with a fixed non-conformal grid. This procedure introduces a stochastic force into the Navier–Stokes equations, and modifies the boundary conditions at the fluid–solid interface. We employ a high-order semi-implicit splitting scheme implemented in the context of a spectral/hp element method in order to discretize the physical domain. The stochastic roughness is treated as a second-order autoregressive process that is represented by a Karhunen– Loeve expansion. A multi-element probabilistic collocation method is employed to solve the resulting stochastic Navier–Stokes equations. This method is applied to simulate external flow past a rough cylinder and internal Stokes flow between two parallel plates with random wall roughness. In the first problem, we develop an analytical solution for the asymptotic behavior of the lift coefficient CL to verify the results. In the second test-case, we compare the mean and the standard deviation of the velocity field to those obtained from a different method called stochastic mapping approach (SMA), developed by Tartakovsky and Xiu (2006).

Coherency in the topology of the instantaneous sensitivity fields of the planar mixing layer was captured using the sensitivity equation method (SEM). In the SEM approach, the partial differential equations governing the evolution of the sensitivity coefficients are derived, discretized, and solved directly, in the present work, using an unsteady finite-volume-based fractional-step algorithm. This allows the investigation of parameter-dependence without performing parametric studies. The present results, from numerical simulations run at Reδ0=200 and Pr=0.71, provide a means to examine how and to what extent perturbations in the Reynolds/Prandtl number locally alter the structure of the flow. Specifically, a two-blade pattern appears as a dominant feature in the sensitivity solution and highlights the physical mechanism leading to vortex thickness growth and enhanced molecular mixing with increasing Reδ0 and Pr. An expression describing the sensitivity of vortex thickness to changes in Reδ0 is also derived and validated using the concept of “nearby” flows.

An analytical solution for the isentropic expansion of subsonic-supersonic two-phase steam flow is provided in this paper. The results are supported by our earlier numerical computations, and available experimental values of others (1973). The two-phase mixture is assumed to undergo an isentropic expansion. However the entropy of the vapor portion of the two-phase mixture increases as the released latent heat of the condensate, reversibly moves toward the vapor, and enhances its entropy, as well as the so called “local stagnation” temperature of the vapor. The mass flow rate of the vapor portion of the two-phase mixture is obtained based on the “local stagnation” conditions, and then corrected to give the mixture mass flow rate. Most of the attempts have been made to develop the solution in a manner similar to those of the ideal gases.