<p>Many numerical approaches exist to solving models of electrical activity in the heart. These models consist of a system of stiff nonlinear ordinary differential equations for the voltage and other variables governing channels, with the voltage coupled to a diffusion term. In this work, we propose a new algorithm that uses two common discretization methods, operator splitting and finite elements. Additionally, we incorporate a temporal integration process known as spectral deferred correction. Using these approaches,</p><p>we construct a numerical method that can achieve arbitrarily high order in both space and time in order to resolve important features of the models, while gaining accuracy and efficiency over lower order schemes.</p><p>Our algorithm employs an operator splitting technique, dividing the reaction-diffusion systems from the models into their constituent parts. </p><p>We integrate both the reaction and diffusion pieces via an implicit Euler method. We reduce the temporal and splitting errors by using a spectral deferred correction method, raising the temporal order and accuracy of the scheme with each correction iteration.</p><p> </p><p>Our algorithm also uses continuous piecewise polynomials of high order on rectangular elements as our finite element approximation. This approximation improves the spatial discretization error over the piecewise linear polynomials typically used, especially when the spatial mesh is refined. </p><p>As part of these thesis work, we also present numerical simulations using our algorithm of one of the cardiac models mentioned, the Two-Current Model. We demonstrate the efficiency, accuracy and convergence rates of our numerical scheme by using mesh refinement studies and comparison of accuracy versus computational time. We conclude with a discussion of how our algorithm can be applied to more realistic models of cardiac electrical activity.</p> / Dissertation

This thesis reports the methods to simulate flows withcomplex boundary such as orifice flow. The method is forgeneral purposes so that it has been tested on different flowsincluding orifice flow. Also it contains a chapter about theexperiment of orifice flow. Higher-order precision interpolation schemes are used inumerical simulation to improve prediction at acceptable gridrefinement. Because higher-order schemes cause instability inconvection-diffusion problems or involve a large computationalkernel, they are implemented with deferred correction method. Alower-order scheme such as upwind numerical scheme is used tomake preliminary guess. A deferred (defect) correction term isadded to maintain precision. This avoids the conflict betweenprecision order and implementation difficulty. The authorproposes a shifting between upwind scheme and centraldifference scheme for the preliminary guess. This has beenproven to improve convergence while higher order schemes havewider range of stability. Non-orthogonal grid is a necessity for complex flow. Usuallyone can map coordinate of such a grid to a transformed domainwhere the grid is regular. The cost is that differentialequations get much more complex form. If calculated directly innon-orthogonal grid, the equations keep simple forms. However,it is difficult to make interpolation in a non-orthogonal grid.Three methods can be used: local correction, shape function andcurvilinear interpolation. The local correction method cannotinsure second-order precision. The shape function method uses alarge computational molecule. The curvilinear interpolationthis author proposes imports the advantage of coordinatetransformation method: easy to do interpolation. A coordinatesystem staggered half control volume used in the coordinatetransformation method is used as accessory to deriveinterpolation schemes. The calculation in physical domain withnon-orthogonal grid becomes as easy as that in a Cartesianorthogonal grid. The author applies this method to calculate turbulentorifice flow. The usual under-prediction of eddy length isimproved with the ULTRA-QUICK scheme to reflect the highgradients in orifice flow. In the last chapter, the author quantifies hydraulicabruptness to describe orifice geometry. The abruptness canhelp engineers to interpolate existing data to a new orifice,which saves detailed experiments

<p>This thesis reports the methods to simulate flows withcomplex boundary such as orifice flow. The method is forgeneral purposes so that it has been tested on different flowsincluding orifice flow. Also it contains a chapter about theexperiment of orifice flow.</p><p>Higher-order precision interpolation schemes are used inumerical simulation to improve prediction at acceptable gridrefinement. Because higher-order schemes cause instability inconvection-diffusion problems or involve a large computationalkernel, they are implemented with deferred correction method. Alower-order scheme such as upwind numerical scheme is used tomake preliminary guess. A deferred (defect) correction term isadded to maintain precision. This avoids the conflict betweenprecision order and implementation difficulty. The authorproposes a shifting between upwind scheme and centraldifference scheme for the preliminary guess. This has beenproven to improve convergence while higher order schemes havewider range of stability.</p><p>Non-orthogonal grid is a necessity for complex flow. Usuallyone can map coordinate of such a grid to a transformed domainwhere the grid is regular. The cost is that differentialequations get much more complex form. If calculated directly innon-orthogonal grid, the equations keep simple forms. However,it is difficult to make interpolation in a non-orthogonal grid.Three methods can be used: local correction, shape function andcurvilinear interpolation. The local correction method cannotinsure second-order precision. The shape function method uses alarge computational molecule. The curvilinear interpolationthis author proposes imports the advantage of coordinatetransformation method: easy to do interpolation. A coordinatesystem staggered half control volume used in the coordinatetransformation method is used as accessory to deriveinterpolation schemes. The calculation in physical domain withnon-orthogonal grid becomes as easy as that in a Cartesianorthogonal grid.</p><p>The author applies this method to calculate turbulentorifice flow. The usual under-prediction of eddy length isimproved with the ULTRA-QUICK scheme to reflect the highgradients in orifice flow.</p><p>In the last chapter, the author quantifies hydraulicabruptness to describe orifice geometry. The abruptness canhelp engineers to interpolate existing data to a new orifice,which saves detailed experiments</p>

In this thesis, high order accurate discretization schemes for partial differential equations are investigated. In the first paper, the linearized two-dimensional Navier-Stokes equations are considered. A special formulation of the boundary conditions is used and estimates for the solution to the continuous problem in terms of the boundary conditions are derived using a normal mode analysis. Similar estimates are achieved for the discretized equations. For the discretization, a second order finite difference scheme on a staggered mesh is used. In Paper II, the analysis for the second order scheme is used to develop a fourth order scheme for the fully nonlinear Navier-Stokes equations. The fully nonlinear incompressible Navier-Stokes equations in two space dimensions are considered on an orthogonal curvilinear grid. Numerical tests are performed with a fourth order accurate Padé type spatial finite difference scheme and a semi-implicit BDF2 scheme in time. In Papers III-V, a class of high order accurate time-discretization schemes based on the deferred correction principle is investigated. The deferred correction principle is based on iteratively eliminating lower order terms in the local truncation error, using previously calculated solutions, in each iteration obtaining more accurate solutions. It is proven that the schemes are unconditionally stable and stability estimates are given using the energy method. Error estimates and smoothness requirements are derived. Special attention is given to the implementation of the boundary conditions for PDE. The scheme is applied to a series of numerical problems, confirming the theoretical results. In the sixth paper, a time-compact fourth order accurate time discretization for the one- and two-dimensional wave equation is considered. Unconditional stability is established and fourth order accuracy is numerically verified. The scheme is applied to a two-dimensional wave propagation problem with discontinuous coefficients.

La simulation numérique de systèmes d’équations différentielles raides ordinaires ou algébriques est devenue partie intégrante dans le processus de conception des systèmes mécaniques à dynamiques complexes. L’objet de ce travail est de développer des méthodes numériques pour réduire les temps de calcul par le parallélisme en suivant deux axes : interne à l’intégrateur numérique, et au niveau de la décomposition de l’intervalle de temps. Nous montrons l’efficacité limitée au nombre d’étapes de la parallélisation à travers les méthodes de Runge-Kutta et DIMSIM. Nous développons alors une méthodologie pour appliquer le complément de Schur sur le système linéarisé intervenant dans les intégrateurs par l’introduction d’un masque de dépendance construit automatiquement lors de la mise en équations du modèle. Finalement, nous étendons le complément de Schur aux méthodes de type "Krylov Matrix Free". La décomposition en temps est d’abord vue par la résolution globale des pas de temps dont nous traitons la parallélisation du solveur non-linéaire (point fixe, Newton-Krylov et accélération de Steffensen). Nous introduisons les méthodes de tirs à deux niveaux, comme Parareal et Pita dont nous redéfinissons les finesses de grilles pour résoudre les problèmes raides pour lesquels leur efficacité parallèle est limitée. Les estimateurs de l’erreur globale, nous permettent de construire une extension parallèle de l’extrapolation de Richardson pour remplacer le premier niveau de calcul. Et nous proposons une parallélisation de la méthode de correction du résidu. / This PhD Thesis deals with the development of parallel numerical methods for solving Ordinary and Algebraic Differential Equations. ODE and DAE are commonly arising when modeling complex dynamical phenomena. We first show that the parallelization across the method is limited by the number of stages of the RK method or DIMSIM. We introduce the Schur complement into the linearised linear system of time integrators. An automatic framework is given to build a mask defining the relationships between the variables. Then the Schur complement is coupled with Jacobian Free Newton-Krylov methods. As time decomposition, global time steps resolutions can be solved by parallel nonlinear solvers (such as fixed point, Newton and Steffensen acceleration). Two steps time decomposition (Parareal, Pita,...) are developed with a new definition of their grids to solved stiff problems. Global error estimates, especially the Richardson extrapolation, are used to compute a good approximation for the second grid. Finally we propose a parallel deferred correction