On Compact Finite Difference Schemes With Applications To Moving Boundary Problems

Compact finite differences are introduced with the purpose of developing compact methods of higher order for the numerical solution of ordinary and elliptic partial differential equations.;The notion of poisedness of a compact finite difference is introduced. It is shown that if the incidence matrix of the underlying interpolation problem contains no odd unsupported sequences then the Polya conditions are necessary and sufficient for poisedness.;A Pade Operator method is used to construct compact formulae valid for uniform three point grids. A second Function-Theoretic method extends compact formulae to variably-spaced three point grids with no deterioration in the order of the truncation error.;A new fourth order compact method (CI4) leading to matrix systems with block tridiagonal structure, is applied to boundary value problems associated with second order ordinary differential equations. Numerical experiments with both linear and nonlinear problems and on uniform and nonuniform grids indicate rates of convergence of four.;An application is considered to the time-dependent one-dimensional nonlinear Burgers\u27 equation in which an initial sinusoidal disturbance develops a very sharp boundary layer. It is found that the CI4 method, with a small number of points placed on a highly stretched grid, is capable of accurately resolving the boundary layer.;A new method (LCM) based on local polynomial collocation and Gauss-type quadrature and leading to matrix systems with block tridiagonal structure, is used to generate high order compact methods for ordinary differential equations. A tenth order method is shown to be considerably more efficient than the CI4 method.;A new fourth order compact method, based on the CI4 method, is developed for the solution, on variable grids, of two-dimensional, time independent elliptic partial differential equations. The method is applied to the ill-posed problem of calculating the interface in receding Hele-Shaw flow. Comparisons with exact solutions indicate that the numerical method behaves as expected for early times.;Finally, in an application to the simulation of contaminant transport within a porous medium under an evolving free surface, new fourth order explicit compact expressions for mixed derivatives are developed

A hierarchy for modeling high speed propulsion systems

General research efforts on reduced order propulsion models for control systems design are overviewed. Methods for modeling high speed propulsion systems are discussed including internal flow propulsion systems that do not contain rotating machinery, such as inlets, ramjets, and scramjets. The discussion is separated into four areas: (1) computational fluid dynamics models for the entire nonlinear system or high order nonlinear models; (2) high order linearized models derived from fundamental physics; (3) low order linear models obtained from the other high order models; and (4) low order nonlinear models (order here refers to the number of dynamic states). Included in the discussion are any special considerations based on the relevant control system designs. The methods discussed are for the quasi-one-dimensional Euler equations of gasdynamic flow. The essential nonlinear features represented are large amplitude nonlinear waves, including moving normal shocks, hammershocks, simple subsonic combustion via heat addition, temperature dependent gases, detonations, and thermal choking. The report also contains a comprehensive list of papers and theses generated by this grant

Solving reaction-diffusion equations 10 times faster

The most popular numerical method for solving systems of reaction-diffusion equations continues to be a low order finite-difference scheme coupled with low order Euler time stepping. This paper extends previous 1D work and reports experiments that show that with high--order methods one can speed up such simulations for 2D and 3D problems by factors of 10--100. A short MATLAB code (2/3D) that can serve as a template is included.\ud \ud This work was supported by the Engineering and Physical Sciences Research Council (UK) and by the MathWorks, Inc

On the relevance of the dam break problem in the context of nonlinear shallow water equations

The classical dam break problem has become the de facto standard in validating the Nonlinear Shallow Water Equations (NSWE) solvers. Moreover, the NSWE are widely used for flooding simulations. While applied mathematics community is essentially focused on developing new numerical schemes, we tried to examine the validity of the mathematical model under consideration. The main purpose of this study is to check the pertinence of the NSWE for flooding processes. From the mathematical point of view, the answer is not obvious since all derivation procedures assumes the total water depth positivity. We performed a comparison between the two-fluid Navier-Stokes simulations and the NSWE solved analytically and numerically. Several conclusions are drawn out and perspectives for future research are outlined.Comment: 20 pages, 15 figures. Accepted to Discrete and Continuous Dynamical Systems. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutyk

Theory, Computation, and Modeling of Cancerous Systems

This dissertation focuses on three projects. In Chapter 1, we derive and implement the compact implicit integration factor method for numerically solving partial differential equations. In Chapters 2 and 3, we generalize and analyze a mathematical model for the nonlinear growth kinetics of breast cancer stem cells. And in Chapter 4, we develop a novel mathematical model for the HER2 signaling pathway to understand and predict breast cancer treatment. Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential equations. Implicit integration method (IIF) method along with its compact form (cIIF), which treats spatial derivatives exactly and reaction terms implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reaction and spatial derivatives. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. The compact representation for IIF (cIIF) was introduced to save the computational cost and storage for this purpose. Another challenge is finding the matrix of high order space discretization, especially near the boundaries. In Chapter 1, we extend IIF method to high order discretization for spatial derivatives through an example of reaction diffusion equation with fourth order accuracy, while the computational cost and storage are similar to the general second order cIIF method. The method can also be efficiently applied to deal with other types of partial differential equations with both homogeneous and inhomogeneous boundary conditions. Direct numerical simulations demonstrate the efficiency and accuracy of the approach. Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to aid in designing experiments to develop novel therapeutic strategies against cancer. In Chapter 2, by using theory of functional and ordinary differential equations, we study the existence and stability of non-linear growth kinetics of breast cancer stem cells. First we provide a sufficient condition for the existence and uniqueness of the solution for non-linear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a non-trivial steady state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions which allow for this criteria to be satisfied. Next we apply these theorems to a special case of the system of functional differential equations that has been used to model non-linear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are a highly tumorigenic cell type found in developmentally diverse tumors that are believed to be resistant to standard chemotherapeutic drugs and responsible for tumor recurrence. Thus understanding the tumor growth kinetics is critical for developing novel strategies for cancer treatment. In Chapter 3, the moment stability of nonlinear stochastic systems of breast cancer stem cells with time-delays is investigated. First, based on the technique of the variation- of-constants formula, we obtain the second order moment equations for the nonlinear stochastic systems of breast cancer stem cells with time-delays. By the comparison principle along with the established moment equations, we can get the comparative systems of the nonlinear stochastic systems of breast cancer stem cells with time-delays. Then moment stability theorems are established for the systems with the stability properties for the comparative systems. Based on the linear matrix inequality (LMI) technique, we next obtain a criteria for the exponential stability in mean square of the nonlinear stochastic systems for the dynamics of breast cancer stem cells with time-delays. Finally, some numerical examples are presented to illustrate the efficiency of the results. Over-expression of human epidermal growth factor receptor 2 (HER2) plays a role in regulation of cancer stem cell (CSC) population in breast cancer. Current cancer therapy includes drugs that block HER2, however, patients can develop anti-HER2 drug resistance. Downstream of HER2 is nuclear factor κB (NFκB). The aberrant regulation of NFκB leads to cancer growth, which makes it a promising target for cancer therapy, especially for those who have developed resistance to anti-HER2 treatment. In Chapter 4 we develop a novel mathematical model that represents the dynamics of the HER2 signaling pathway. By integrating experimental data with model simulations, we discover that interleukin-1 (IL1), which is downstream of HER2, is responsible for NFκB activation. We perform global sensitivity analysis on the model to identify key reactions. Our modeling effort shows that IL1 is critical in NFκB regulation, especially in the absence of HER2, making it a potential target in treating breast cancer for patients who have developed resistance to anti-HER2 drugs

Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations

This paper introduces tensorial calculus techniques in the framework of Proper Orthogonal Decomposition (POD) to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree $p$. Such nonlinear terms have an on-line complexity of $\mathcal{O}(k^{p+1})$, where $k$ is the dimension of POD basis, and therefore is independent of full space dimension. However it is efficient only for quadratic nonlinear terms since for higher nonlinearities standard POD proves to be less time consuming once the POD basis dimension $k$ is increased. Numerical experiments are carried out with a two dimensional shallow water equation (SWE) test problem to compare the performance of tensorial POD, standard POD, and POD/Discrete Empirical Interpolation Method (DEIM). Numerical results show that tensorial POD decreases by $76\times$ times the computational cost of the on-line stage of standard POD for configurations using more than $300,000$ model variables. The tensorial POD SWE model was only $2-8\times$ slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM shallow water equation model to compute its off-line stage faster than the standard and tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table

Control of Integrable Hamiltonian Systems and Degenerate Bifurcations

We discuss control of low-dimensional systems which, when uncontrolled, are integrable in the Hamiltonian sense. The controller targets an exact solution of the system in a region where the uncontrolled dynamics has invariant tori. Both dissipative and conservative controllers are considered. We show that the shear flow structure of the undriven system causes a Takens-Bogdanov birfurcation to occur when control is applied. This implies extreme noise sensitivity. We then consider an example of these results using the driven nonlinear Schrodinger equation.Comment: 25 pages, 11 figures, resubmitted to Physical Review E March 2004 (originally submitted June 2003), added content and reference

Developing Clean Technology through Approximate Solutions of Mathematical Models

In this paper, the role of mathematical modeling in the development of clean technology has been considered. One method each for obtaining approximate solutions of mathematical models by ordinary differential equations and partial differential equations respectively arising from the modeling of systems and physical phenomena has been considered. The construction of continuous hybrid methods for the numerical approximation of the solutions of initial value problems of ordinary differential equations as well as homotopy analysis method, an approximate analytical method, for the solution of nonlinear partial differential equations are discussed