In this thesis, we make use of numerical schemes in order to solve Fisher’s and FitzHugh-Nagumo equations with specified initial conditions. The thesis is made up of six chapters. Chapter 1 gives some literatures on partial differential equations and chapter 2 provides some concepts on finite difference methods, nonstandard finite difference methods and their proper-ties, reaction-diffusion equations and singularly perturbed equations. In chapter 3, we obtain the numerical solution of Fisher’s equation when the coefficient of diffu-sion term is much smaller than the coefficient of reaction (Li et al., 1998). Li et al. (1998) used the Moving Mesh Partial Differential Equation (MMPDE) method to solve a scaled Fisher’s equation with coefficient of reaction being 104 and coefficient of diffusion equal to one and the initial condition consisted of an exponential function. The problem considered is quite challeng-ing and the results obtained by Li et al. (1998) are not accurate due to the fact that MMPDE is based on familiar arc-length or curvature monitor function. Qiu and Sloan (1998) constructed a suitable monitor function called modified monitor function and used it with the Moving Mesh Differential Algebraic Equation (MMDAE) method in order to solve the same problem as Li et al. (1998) and better result were obtained. However, each problem has its own choice of monitor function which makes the choice of the monitor function an open question. We use the Forward in Time Central Space (FTCS) scheme and the Nonstandard Finite Difference (NSFD) to solve the scaled Fisher’s equation and we find that the temporal step size must be very small in order to obtain accurate results and comparable to Qiu and Sloan (1998). This causes the computational time to be long if the domain is large. We use two techniques to modify these two schemes either by introducing artificial viscosity or using the approach of Ruxun et al. (1999). These techniques are efficient and give accurate results with a larger temporal step size. We prove that these four methods are consistent with the partial differential equation and we also obtain the region of stability. Chapter 4 is an improvement and extension of the work from Namjoo and Zibaei (2018) whereby the standard FitzHugh-Nagumo equation with specified initial and boundary conditions is solved. Namjoo and Zibaei (2018) constructed two versions of nonstandard finite difference (NSFD1, NSFD2) and also derived two schemes (one explicit and the other implicit) constructed from the exact solution. However, they presented results using the nonstandard finite difference schemes only. We showed that one of the nonstandard finite difference schemes (NSFD1) has convergence issues and we obtain an improvement for NSFD1 which we call NSFD3. We per-form a stability analysis of the schemes constructed from the exact solution and found that the explicit scheme is not stable for this problem. We study some properties of the five methods (NSFD1, NSFD2, NSFD3, two schemes obtained using the exact solution) such as stability, positivity and boundedness. The performance of the five methods is compared by computing L1, L∞ errors and the rate of convergence for two values of the threshold of Affect effect, γ namely; 0.001 and 0.5 for small and large spatial domains at time, T = 1.0. Tests on rate of convergence are important here as we are dealing with nonlinear partial differential equations and therefore the Lax-Equivalence theorem cannot be used. In chapter 5, we consider FitzHugh-Nagumo equation with the parameter β referred to as in-trinsic growth rate. We chose a numerical experiment which is quite challenging for simulation due to shock-like profiles. We construct four versions of nonstandard finite difference schemes and compared the performance by computing L1, L∞ errors, rate of convergence with respect to time and CPU time at given time, T = 0.5 using three values of the intrinsic growth rate, β namely; β = 0.5, 1.0, 2.0. Chapter 6 highlights the salient features of this work. / Thesis (PhD)--University of Pretoria, 2020. / South African DST/NRF SARChI / Mathematics and Applied Mathematics / PhD / Unrestricted

Un nouveau code de simulation numérique pour la propagation des ondes de choc acoustiques dans des géométries complexes a été développé. Le point de départ a été la méthode de Galerkin discontinu qui utilise des maillages non structurés (ici des éléments triangulaires), particulièrement adaptés aux géométries complexes. Cependant, cette discrétisation conduit à l'apparition d'oscillation de Gibbs. Pour pallier ce problème, nous avons choisi d'introduire de la viscosité artificielle au voisinage des chocs. Cela a nécessité le développement de trois outils originaux : (i) un nouveau détecteur de choc sensible aux ondes de chocs acoustiques sur des maillages non structurés, (ii) un nouveau terme de viscosité artificielle dans les équations de l'acoustique non linéaire défini élément par élément et (iii) un nouveau terme permettant de régler le niveau de viscosité locale à partir du raidissement des fronts d'onde. Le code de calcul a été utilisé pour étudier deux configurations différentes. La première concerne la réflexion d'ondes de choc acoustiques sur des surfaces rigides. Différents régimes de réflexion ont alors été observés allant, de la réflexion classique de Snell Descartes jusqu'à celui dit de réflexion faible de Von Neumann. La deuxième configuration était consacrée à la focalisation d'ondes de choc acoustiques produites par un transducteur à haute intensité (comme ceux utilisés en HIFU). Un soin particulier a été pris pour étudier le calcul de l'intensité et pour étudier l'interaction entre les ondes de choc et des obstacles placés dans la région du foyer. / A new numerical solver for the propagation of acoustical shock waves in complex geometry has been developed. This is done starting from the discontinuous Galerkin method. This method is based on unstructured mesh (triangular elements here), and so, naturally it is well-adapted for complex geometries. Nevertheless, the discretization induces Gibbs oscillations. To manage this problem, we choose to introduce some artificial viscosity only in the vicinity of the shocks. This necessitates the development of three original tools. First of all, a new shock sensor for unstructured mesh sensitive to acoustical shock waves has been designed. It senses where the local artificial viscosity has to be introduced thanks to a reformulation of a new element centred smooth artificial viscosity term in the equations. Finally, the amount of viscosity is computed by the introduction of an original notion of gradient factor linked to the steepening of the waveform. The numerical solver has been used to investigate two different physical situations. The first one is the nonlinear reflection of acoustical shock waves on rigid surfaces. Different regimes of reflection have been observed ranging from the linear Snell Descartes reflection to the weak von Neumann case. The second configuration deals with the focusing of shock waves produced by high intensity transducers (like in HIFU). Special attention has been given to the careful computation of intensity and to the interaction between the shock waves and obstacles in the region of the focus.

Chorin’s method, Incremental Pressure Correction Scheme (IPCS) and Crank-Nicolson’s method (CN) are three numerical methods that were investigated in this study. These methods were here used for solving the incompressible Navier-Stokes equations, which describe the motion of an incompressible fluid, in three different benchmark problems. The methods were stabilized using residual based artificial viscosity, which was introduced to avoid instability. The methods were compared in terms of accuracy and computational time. Furthermore, a theoretical study of adaptivity was made, based on an a posteriori error estimate and an adjoint problem. The implementation of the adaptivity is left for future studies. In this study we consider the following three well-known benchmark problems: laminar 2D flow around a cylinder, Taylor-Green vortex and lid-driven cavity problem. The difference of the computational time for the three methods were in general relatively small and differed depending on which problem that was investigated. Furthermore the accuracy of the methods also differed in the benchmark problems, but in general Crank-Nicolson’s method gave less accurate results. Moreover the stabilization technique worked well when the kinematic viscosity of the fluid was relatively low, since it managed to stabilize the numerical methods. In general the solution was affected in a negative way when the problem could be solved without stabilization for higher viscosities.

This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p > 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization.

L’objectif de ce travail de thèse est de proposer la Méthodes des éléments finis de Galerkin discontinus (DGFEM) à la discrétisation des équations compressibles de Navier-Stokes. Plusieurs challenges font l’objet de ce travail. Le premier aspect a consisté à montrer l’ordre de convergence optimal de la méthode DGFEM en utilisant les polynômes d’interpolation d’ordre élevé. Le deuxième aspect concerne l’implémentation de méthodes de ‘‘shock-catpuring’’ comme les limiteurs de pentes et les méthodes de viscosité artificielle pour supprimer les oscillations numériques engendrées par l’ordre élevé (lorsque des polynômes d’interpolation de degré p>0 sont utilisés) dans les écoulements transsoniques et supersoniques. Ensuite nous avons implémenté des estimateurs d’erreur a posteriori et des procédures d ’adaptation de maillages qui permettent d’augmenter la précision de la solution et la vitesse de convergence afin d’obtenir un gain de temps considérable. Finalement, nous avons montré la capacité de la méthode DG à donner des résultats corrects à faibles nombres de Mach. Lorsque le nombre de Mach est petit pour les écoulements compressibles à la limite de l’incompressible, la solution souffre généralement de convergence et de précision. Pour pallier ce problème généralement on procède au préconditionnement qui modifie les équations d’Euler. Dans notre cas, les équations ne sont pas modifiées. Dans ce travail, nous montrons la précision et la robustesse de méthode DG proposée avec un schéma en temps implicite de second ordre et des conditions de bords adéquats. / The aim of this thesis is to deal with compressible Navier-Stokes flows discretized by Discontinuous Galerkin Finite Elements Methods. Several aspects has been considered. One is to show the optimal convergence of the DGFEM method when using high order polynomial. Second is to design shock-capturing methods such as slope limiters and artificial viscosity to suppress numerical oscillation occurring when p>0 schemes are used. Third aspect is to design an a posteriori error estimator for adaptive mesh refinement in order to optimize the mesh in the computational domain. And finally, we want to show the accuracy and the robustness of the DG method implemented when we reach very low mach numbers. Usually when simulating compressible flows at very low mach numbers at the limit of incompressible flows, there occurs many kind of problems such as accuracy and convergence of the solution. To be able to run low Mach number problems, there exists solution like preconditioning. This method usually modifies the Euler. Here the Euler equations are not modified and with a robust time scheme and good boundary conditions imposed one can have efficient and accurate results.