Amit Sharma (Advisor), Mohamed Sulman (Advisor), Brent Foy (Committee Member), Thomas Skinner (Committee Member)
Doctor of Philosophy (PhD)
In this thesis, we develop efficient numerical solvers for nonlinear systems of partial differential equations (PDEs). These systems of PDEs concern two different sets of physical problems. The first set includes chemotaxis models such as Keller-Segel models and cancer cell invasion models. Solutions of these models are observed to experience the blow-up phenomenon or the development of sharp and spiky features. Therefore, efficient and accurate numerical techniques must be employed in order to capture the solutions' behaviors. For this research, we design efficient solvers for these systems in the one and two spatial dimensions. In particular, we plan to apply adaptive moving mesh methods in which the mesh points are continuously redistributed by a coordinate transformation from the computational domain to the physical domain so that the grid nodes are concentrated in regions of large solution variations in the physical domain. The second system is the system of nonlinear PDEs that govern the plasma modulation instability of wave collapse. It is known that the nonlinear interaction of lower-hybrid wave with a much lower frequency plasma perturbation leads to the development of modulation instability which causes oscillations of electric field and formations of cavitons. Cavitons, which are regions where plasma's density is observed to be decreased, collapse in finite time and during this period, the trapped energy of electric field oscillations is dissipated and electrons are heated up. Direct experiment observations of collapse phenomenon of cavitons can be difficult. Therefore, numerical simulations are desired. To overcome the above challenges and achieve the thesis' goals, we first study the basic mechanisms of the adaptive moving mesh methods by implementing adaptive grid methods using finite difference and finite element discretization. We then apply them for well known toy problems e.g., Burgers' equations. On the other hand, we re-implement the pseudo-spectral method and we also apply the method to compute solutions to simple problems e.g., solving Poisson problems with periodic boundary conditions. Once the methods are tested with toy examples, we are ready to apply them to obtain numerical solutions to the nonlinear diffusion-reaction-chemotaxis models (cancer invasion models and Keller-Segel models) and the system of nonlinear equations that govern the modulation instability. In particular, we apply adaptive moving mesh methods for nonlinear diffusion-reaction-chemotaxis models. These model equations are discretized using finite difference (FD) and/or finite element (FE) methods. Positivity preserving schemes are used for the spatial discretization of these chemotaxis models to ensure that the physical solutions remain positive at all time levels. Numerical experiments are performed to demonstrate the performance of the adaptive mesh methods. Our numerical results show that the proposed moving mesh methods reduce the computational cost while improving the accuracy of the computed solutions when comparing to uniform grid methods. Meanwhile, we follow Shapiro's approach in his 1993 paper to further test our pseudo-spectral solver for solving the governing system of plasma modulation instability in two and three spatial dimensions. The pseudo-spectral method utilizes the computation of the fast Fourier transform (FFT) which is done by using FFTW library. High order time integration techniques are applied to calculate solutions at the a time. We further extend the pseudo-spectral method to a highly scalable solver by implementing it in parallel using Message Passing Interface (MPI). Our MPI code allows us to speed up the computations especially in the three dimensional problem. Next, we use our solvers to numerically study the effect of plasma's shear velocity on the wave collapse phenomenon. Finally, we implement a solver that uses adaptive moving mesh finite difference method for the system of PDEs that govern the modulation instability phenomenon in the two dimensional case.
Department or Program
Department of Mathematics and Statistics
Year Degree Awarded
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