An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator

Ensemble-based implicit sampling for Bayesian inverse problems with non-Gaussian priors

In the paper, we develop an ensemble-based implicit sampling method for Bayesian inverse problems. For Bayesian inference, the iterative ensemble smoother (IES) and implicit sampling are integrated to obtain importance ensemble samples, which build an importance density. The proposed method shares a similar idea to importance sampling. IES is used to approximate mean and covariance of a posterior distribution. This provides the MAP point and the inverse of Hessian matrix, which are necessary to construct the implicit map in implicit sampling. The importance samples are generated by the implicit map and the corresponding weights are the ratio between the importance density and posterior density. In the proposed method, we use the ensemble samples of IES to find the optimization solution of likelihood function and the inverse of Hessian matrix. This approach avoids the explicit computation for Jacobian matrix and Hessian matrix, which are very computationally expensive in high dimension spaces. To treat non-Gaussian models, discrete cosine transform and Gaussian mixture model are used to characterize the non-Gaussian priors. The ensemble-based implicit sampling method is extended to the non-Gaussian priors for exploring the posterior of unknowns in inverse problems. The proposed method is used for each individual Gaussian model in the Gaussian mixture model. The proposed approach substantially improves the applicability of implicit sampling method. A few numerical examples are presented to demonstrate the efficacy of the proposed method with applications of inverse problems for subsurface flow problems and anomalous diffusion models in porous media.Comment: 27 page

Bilateral tempered fractional derivatives

Publisher Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland.The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it cannot be considered as a derivative.publishersversionpublishe

Simulation of 2-dimensional viscous flow through cascades using a semi-elliptic analysis and hybrid C-H grids

A semi-elliptic formulation, termed the interacting parabolized Navier-Stokes (IPNS) formulation, is developed for the analysis of a class of subsonic viscous flows for which streamwise diffusion is neglible but which are significantly influenced by upstream interactions. The IPNS equations are obtained from the Navier-Stokes equations by dropping the streamwise viscous-diffusion terms but retaining upstream influence via the streamwise pressure-gradient. A two-step alternating-direction-explicit numerical scheme is developed to solve these equations. The quasi-linearization and discretization of the equations are carefully examined so that no artificial viscosity is added externally to the scheme. Also, solutions to compressible as well as nearly compressible flows are obtained without any modification either in the analysis or in the solution process. The procedure is applied to constricted channels and cascade passages formed by airfoils of various shapes. These geometries are represented using numerically generated curilinear boundary-oriented coordinates forming an H-grid. A hybrid C-H grid, more appropriate for cascade of airfoils with rounded leading edges, was also developed. Satisfactory results are obtained for flows through cascades of Joukowski airfoils

Efficient numerical schemes for porous media flow

Partial di erential equations (PDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Due to the uncertainty in the input data, stochastic partial di erential equations (SPDEs) have become popular as a modelling tool in the last century. As the exact solutions are unknown, developing e cient numerical methods for simulating PDEs and SPDEs is a very important while challenging research topic. In this thesis we develop e cient numerical schemes for deterministic and stochastic porous media ows. More schemes are based on the computing of the matrix exponential functions of the non diagonal matrices, we use new e cient techniques: the real fast L eja points and the Krylov subspace techniques. For the deterministic ow and transport problem, we consider two deterministic exponential integrator schemes: the exponential time di erential stepping of order one (ETD1) and the exponential Euler midpoint (EEM) with nite volume method for discretization in space. We give the time and space convergence proof for the ETD1 scheme and illustrate with simulations in two and three dimensions that the exponential integrators are e - cient and accurate for advection dominated deterministic transport ow in heterogeneous anisotropic porous media compared to standard semi implicit and implicit schemes. For the stochastic ow and transport problem, we consider the general parabolic SPDEs in a Hilbert space, using the nite element method for discretization in space (although nite di erence or nite volume can be used as well). We use a linear functional of the noise and the standard Brownian increments to develop and give convergence proofs of three new e cient and accurate schemes for additive noise, one called the modi ed semi{ implicit Euler-Maruyama scheme and two stochastic exponential integrator schemes, and two stochastic exponential integrator schemes for multiplicative and additive noise. The schemes are applied to two dimensional ow and transport

Numerical Methodologies for Non-Equilibrium Plasma Modelling

The development of reliable numerical tools for the simulation of non-equilibrium plasma devices is a fundamental requirement the technological progress. The main challenge in this context is to adequately represent multiple physical phenomena that take place over different temporal and spatial scales, while retaining reasonable computational performances. In the first part of the work, a fluid methodology in which electrons are modelled using the Boltzmann relation is developed as an alternative to Full Drift-Diffusion models. The proposed Boltzmann Drift-Diffusion methodology allows to limit the drift-diffusion approach to the ionic species, granting substantial savings in terms of computational performances. Both methodologies are applied to the 1D/2D simulation of a volumetric Dielectric Barrier Discharge reactor, operating with air at atmospheric pressure. A semi-implicit numerical technique for the integration of plasma kinetic processes is presented and numerically validated against a well established implicit methodology. The Boltzmann Drift-Diffusion and Full Drift-Diffusion approaches are compared. The obtained results are validated against experimental measurements of the deposited surface charge on to the dielectric layers covering the electrodes. In the second part of the work, a hybrid fluid/Particle-In-Cell approach is employed to model a miniaturized annular Hall thruster for space propulsion. The results yielded by two different treatments of the electron transport mechanism inside and outside the thruster channel are compared to macroscopic and microscopic physical information obtained through experimental measurements. These latter are then used to infer the anomalous transport collision frequency along the axis of the thruster. The obtained efficiency of the two chemical kinetic channels for the doubly charged ions production is discussed and correlated with the computed spatial distribution of the species

Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims

Fractional Diffusion Modeling of Electromagnetic Induction in Fractured Rocks

The controlled-source electromagnetic (CSEM) technique is well-established for non-invasive geophysical survey. Due to the strong attenuation of earth materials to electromagnetic signals, the effective depth of most CSEM surveys is restricted to 1-2 km, a zone where pores and fractures over various length scales are highly complicated. Spatial confinement of fluid or electric charge transport by the fractal geometry gives rise to interesting dynamic processes within the pore space and fractures, such as anomalous diffusion. Conventionally, CSEM data are interpreted in terms of a 1-D, 2-D or 3-D piecewise constant geological structure with uniform conductivity and thickness of each cell. A very fine grid, and hence a lot of computation time, are needed to build and evaluate a model that can explain the Earths actual 3D CSEM response. Good accuracy may not be captured, using the conventional approach, in the presence of multi-scale hierarchical geoelectrical structure. Alternatively, the CSEM response of such structures are easily evaluated if the physics of anomalous diffusion of electromagnetic eddy currents is recognized and cast, for example, in terms of a continuous time random walk. Such a re-formulation leads to a generalization of Maxwell equations containing a fractional order time derivative. The fractional order of the derivative is equivalent to a roughening of the geological medium, introducing multi-scale variations of fractures and heterogeneities in a compact manner. This theory renders CSEM modeling and inversion much more efficient, as only a few model parameters are now required to be fit. However the EM fractional diffusion theory is far from perfect, e.g. the correlation between the roughness of a fracture model with its fracture properties. In this research, I use numerical modeling tool to answer this question and explore if classical piece-wise constant conductivity model can generate a fractional type response. In this thesis, I will review the fundamental theory of traditional CSEM survey technique and the continuous time random walk approach, and review the derivation of the generalized Maxwell equation. More importantly, I propose the finite difference method to discrete the generalized Maxwell equation in 2D and 3D. I explore a classical fractured model response created from the von Karman random media approach. I will show that the von Karman fractured model generates a classical type response which is inconsistent with the fractional diffusion response. It is difficult to generate a classical model numerically that is comparable with the rough natural model