Exponential Integrator Methods for Nonlinear Fractional Reaction-diffusion Models

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this work, we propose an exponential integrator method for nonlinear fractional reaction-diffusion equations. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. Due to these real distinct poles, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second-order convergent; and proven to be robust for problems involving non-smooth/mismatched initial and boundary conditions and steep solution gradients. We examine the stability of the scheme through its amplification factor and plot the boundaries of the stability regions comparative to other second-order FETD schemes. This numerical scheme combined with fractional centered differencing is used for simulating many important nonlinear fractional models in applications. We demonstrate the superiority of our method over competing second order FETD schemes, BDF2 scheme, and IMEX schemes. Our experiments show that the proposed scheme is computationally more efficient (in terms of cpu time). Furthermore, we investigate the trade-off between using fractional centered differencing and matrix transfer technique in discretization of Riesz fractional derivatives. The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative. Furthermore, we present a preliminary result of the application of the M-L RDP approximation to develop a generalized exponetial integrator scheme for time-fractional nonlinear reaction-diffusion equation

SCALAR-SOURCE IDENTIFICATION AND OPTIMAL SENSOR PLACEMENT IN TURBULENT CHANNEL FLOW

The spreading of a released pollutant in a turbulent environment has severe consequences. The ability to identify the unknown source location from remote sensor data is greatly obfuscated by turbulence. This work discusses effective scalar-source localization algorithms in a turbulent channel by exploiting adjoint and ensemble methods, and by utilizing the growing power of high-fidelity simulations. To reconstruct the spatial distribution of the source, a cost functional is defined based on the difference between the true sensor observations and their model predictions. Forward-adjoint simulations provide the gradient of the cost functional to the source distribution, and the source estimation is iteratively updated. When a single sensor is directly downstream, the reconstruction is accurate in the cross-stream directions but elongated in streamwise direction. Using more sensors improves the performance demonstrably. We therefore seek the optimal sensor placement that improves the prediction in streamwise direction, by minimizing the condition number of the Hessian matrix of the cost functional. An iterative approach is adopted that gradually adjusts the sensor(s) while tracking the principal subspace of the Hessian. For a single sensor, the optimal location is near the edge of the scalar plume. This placement distinguishes signals for adjacent sources much more than sensor directly downstream. For fast identification of the source location and intensity, an eigen-ensemble-variational algorithm is formulated, which relies on the left and right singular vectors, or eigen-sources and eigen-measurements of the scalar impulse-response system. The projection of the true source onto an eigen-source is proportional to the projection of the sensor signal onto the corresponding eigen-measurement. The unknown source is identified by minimizing its deviation from this proportionality. We demonstrate effective ways to use an ensemble of trial sources to estimate the pre-requisite eigen-sources and accurately predict the source location. Furthermore, the effect of sensor noise can be evaluated when Gaussian noise is added to the measurement. All together, the developed algorithms provide effective strategies for reconstruction of unknown scalar sources and optimization of sensor networks. The resulting data provide an important benchmark for future research on olfactory search strategies in fully turbulent environments

Numerical methods and tangible interfaces for pollutant dispersion simulation

Dissertação apresentada para obtenção do Grau de Doutor em Engenharia do Ambiente, pela Universidade Nova de Lisboa, Faculdade de Ciências e TecnologiaThe first main objective of this thesis is to reduce numerical errors in advection-diffusion modelling. This is accomplished by presenting DisPar methods, a class of numerical schemes for advection-diffusion or transport problems, based on a particle displacement distribution for Markov processes. The development and analyses of explicit and implicit DisPar formulations applied to one and two dimensional uniform grids are presented. The first explicit method, called DisPar-1, is based on the development of a discrete probability distribution for a particle displacement, whose numerical values are evaluated by analysing average and variance. These two statistical parameters depend on the physical conditions (velocity, dispersion coefficients and flows). The second explicit method,DisPar-k, is an extension of the previous one and it is developed for one and two dimensions. Besides average and variance, this method is also based on a specific number of particle displacement moments. These moments are obtained by the relation between the advection-diffusion and the Fokker-Planck equation, assuming a Gaussian distribution for the particle displacement distribution. The number of particle displacement moments directly affects the spatial accuracy of the method, and it is possible to achieve good results for pure-advection situations. The comparison with other methods showed that the main DisPar disadvantage is the presence of oscillations in the vicinity of step concentration profiles. However, the models that avoid those oscillations generally require complex and expensive computational techniques, and do not perform so well as DisPar in Gaussian plume transport. The application of the 2-D DisPar to the Tagus estuary demonstrates the model capacity of representing mass transport under complex flows. Finally, an implicit version of DisPar is also developed and tested in linear conditions, and similar results were obtained in terms of truncation error and particle transport methods. The second main objective of this thesis, to contribute to modelling cost reduction, is accomplished by presenting TangiTable, a tangible interface for pollutant dispersion simulation composed by a personal computer, a camera, a video projector and a table. In this system, a virtual environment is projected on the table, where the users place objects representing infrastructures that affect the water of an existent river and the air quality. The environment and the pollution dispersion along the river are then projected on the table. TangiTable usability was tested in a public exhibition and the feedback was very positive. Future uses include public participation and collaborative work applications.Fundação para a Ciência e Tecnologia - scholarship contract BD/5064/2001 and the research contract MGS/33998/99-0

Optimum experimental design for parameter estimation with 2D partial differential equation models

In this thesis, we investigate optimization problems with partial differential equation (PDE) constraints. In particular we are concerned with the efficient numerical solution of optimum experimental design (OED) problems for parameter estimation (PE) with PDE models, among them sampling design problems. We consider two dimensional (2D) stationary diffusion advection reaction PDE models, including the challenging case of an advection dominated PDE. For the simulation of the PDE boundary value problem, we utilize discontinuous Galerkin finite element methods and adaptive spatial grid refinement. We solve the optimization problems with derivative-based algorithms. For the optimization algorithms to converge fast and to converge to the ”true“ optimum, we need to provide accurate sensitivities. It is a challenge to evaluate the sensitivities, which correspond to the approximate solution of the primal PDE model and are in this sense consistent. In this thesis we develop efficient and accurate methods for sensitivity generation. We transfer the principle of internal numerical differentiation (IND) from ordinary differential equations (ODE)s to PDEs. That means, we incorporate the sensitivity generation in the solution process. The standard upwind discontinuous Galerkin method is not differentiable. Therefore, we propose a differentiable discontinuous Galerkin method and give a rigorous convergence analysis of it. We develop methods for structure exploitation of the primal and tangential discretization schemes to efficiently generate the sensitivities with automatic differentiation (AD). Furthermore, we establish methods for frozen adaptivity to generate consistent sensitivities. We are especially concerned with frozen spatial grid refinement and the adaptive step number of the linear solver. We implement the developed methods in the software SeafaND-Optimizer, short for structure exploiting and frozen adaptivity numerical differentiation optimizer. It is a software for efficient and accurate simulation, PE and OED with PDE models. We perform numerical case studies for PE and OED problems with advection dominated 2D diffusion advection PDE models. With the structure exploiting techniques developed in this thesis, the example problems are solved with efficient memory usage. Due to the frozen adaptivity methods, we computed efficiently the consistent sensitivities. We test the PE algorithm with different noise levels. We perform a case study with different diffusion coefficients for sequential OED. Finally, we investigate, whether the developed methods are stable under mesh refinements

Modelling the Eddy Pattern in the harbour of Zeebrugge

Hydrodynamic

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis