Numerical Analysis and Scientific Computing

Numerical Solution of Partial Differential Equation Problems

Dr. Ryan I. Fernandes (PI)

  • This project encompasses formulation, convergence analysis and numerical implementation of the alternating direction implicit (ADI) method for solving parabolic problems on convex ADI methods were introduced over 70 decades ago for solving time-dependent problems on rectangles. They have proved to be efficient and are important today as they naturally lend themselves to parallelization. The research envisaged in this project is first of its kind in the literature. We formulate the ADI technique to solve such problems on convex and possibly more complicated regions. Our current focus is on ADI finite differences and will be followed by ADI orthogonal spline collocation methods.

Applications of ADI Techniques to Problems in Physics, Biology and Engineering

Dr. Ryan I. Fernandes (PI)

  • In this project we showcase the ADI technique as a preferred method of choice to be used in applications that arise in many fields, in particular problems from Physics, Biology and Engineering. The finite element Galerkin and orthogonal spline collocation techniques yield an approximate solution over the whole domain (not just at the nodes) with a high degree of accuracy. Together with the efficiency of the ADI technique they should naturally be preferred over other techniques for application problems. Currently, we have formulated a fully discrete ADI Galerkin method to solve Schrӧdinger problems by writing it as a Schrӧdinger type system. Problems of this kind arise in many disciplines such as quantum mechanics, underwater acoustics, plasma physics, and seismology.

Characterization and Modelling of Unconventional Reservoirs

Dr. Ryan I. Fernandes (Co-PI)

  • This is a Master’s thesis project led by Dr. Hadi Belhaj (Petroleum Engineering Dept). The characterization and modeling of an unconventional reservoir are important in order to understand the fluid flow behavior within the nanopores of tight rock formation. The project involves developing a mathematical model of unconventional reservoirs to include various forces (viscous, viscoelastic, capillary, inertia, advection, convection, adsorption and desorption) that are responsible for fluid flow through nanopores in tight rocks. The model will be validated with a numerical study to determine the predominating force(s) that affect the fluid flow. Variation of temperature, pressure and rock properties will show the influence of parameters on fluid flow behavior. This project will help to develop a concept and knowledge base of unconventional reservoir models.

Optimized Schwarz Waveform Relaxation (SWR) Methods for PDEs

Dr. Mohammad Al-Khaleel

  • The so-called Schwarz Waveform Relaxation (SWR) method has been introduced as one of the most efficient parallel methods for solving time dependent problems. It got its name from the fact that in this method the whole spatial domain is decomposed into several subdomains which produce smaller problems, like Schwarz methods, and then on each subdomain a time dependent problem needs to be solved independently but simultaneously for the entire time interval through iterations and hence the method is also of waveform relaxation type. In this project, we consider different PDEs as our model problems and investigate the best performance that one might get using what is called optimized transmission conditions and perform the analysis at a discrete level, at a semi-discrete level and at a continuous level.

Optimized Waveform Relaxation (WR) Methods for Circuit Simulations

Dr. Mohammad Al-Khaleel

  • The WR algorithms have many favorable properties and are extensively studied and investigated in circuit simulations and have been used efficiently to solve the large systems of ODEs obtained from VLSI circuits. One of the most challenging problems for these algorithms is to know what information should be exchanged between the sub-circuits or sub-systems through what is called transmission conditions to obtain the best performance. In this project, we consider different types of VLSI circuits and investigate the best performance that one might get using optimized transmission conditions and perform the analysis at a discrete level and at a continuous level.

Parallel Efficient Algorithms for Large Scale Optimal Control Problems

Dr. Mohammad Al-Khaleel and Dr. Mohamed Kamel Riahi

  • This project is concerned with the design of novel efficient parallel algorithms for large scale optimal control problems. The classical approach for solving optimal control problems is time and memory consuming, in this work we are exploiting parareal-type methods and Waveform Relaxation techniques as well as control decomposition in order to develop robust numerical schemes for optimal control problems.