An Explicit Positivity Preserving Scheme: Application to Biological Model

This paper deals with the construction of nonstandard finite difference method (NFSD) for nonlinear initial value problems modeled by a system of nonlinear ordinary differential equations. The proposed scheme preserves the positivity property as well as the requirement of conservation law and boundedness. In order to illustrate the accuracy of the new scheme, the numerical results compared with the standard ones. Keywords: Positivity, Boundedness, Nonstandard finite difference

A positive and elementary stable nonstandard explicit scheme for a mathematical model of the influenza disease

[EN]In this paper, a nonstandard explicit discretization strategy is considered to construct a new nonstandard finite difference scheme for solving a mathematical model of the influenza disease. The new proposed scheme has some interesting properties such as high accuracy and ease of implementation, as well as some preserving properties of the exact theoretical solution of the SIRC system, like positivity and elementary stability. These characteristics make it suitable for solving efficiently the propose model. We provide some numerical comparisons to illustrate our results

An improvement on the positivity results for 2-stage explicit Runge–Kutta methods

AbstractIn this paper, we investigate the positivity property for a class of 2-stage explicit Runge–Kutta (RK2) methods of order two when applied to the numerical solution of special nonlinear initial value problems (IVPs) for ordinary differential equations (ODEs). We also pay particular attention to monotonicity property. We obtain new results for positivity which are important in practical applications. We provide some numerical examples to illustrate our results

Doctor of Philosophy

dissertationThe Material Point Method (MPM) and the Implicit Continuous-fluid Eulerian method (ICE) have been used to simulate and solve many challenging problems in engineering applications, especially those involving large deformations in materials and multimaterial interactions. These methods were implemented within the Uintah Computational Framework (UCF) to simulate explosions, fires, and other fluids and fluid-structure interaction. For the purpose of knowing if the simulations represent the solutions of the actual mathematical models, it is important to fully understand the accuracy of these methods. At the time this research was initiated, there were hardly any error analysis being done on these two methods, though the range of their applications was impressive. This dissertation undertakes an analysis of the errors in computational properties of MPM and ICE in the context of model problems from compressible gas dynamics which are governed by the one-dimensional Euler system. The analysis for MPM includes the analysis of errors introduced when the information is projected from particles onto the grid and when the particles cross the grid cells. The analysis for ICE includes the analysis of spatial and temporal errors in the method, which can then be used to improve the method's accuracy in both space and time. The implementation of ICE in UCF, which is referred to as Production ICE, does not perform as well as many current methods for compressible flow problems governed by the one-dimensional Euler equations - which we know because the obtained numerical solutions exhibit unphysical oscillations and discrepancies in the shock speeds. By examining different choices in the implementation of ICE in this dissertation, we propose a method to eliminate the discrepancies and suppress the nonphysical oscillations in the numerical solutions of Production ICE - this improved Production ICE method (IMPICE) is extended to solve the multidimensional Euler equations. The discussion of the IMPICE method for multidimensional compressible flow problems includes the method's detailed implementation and embedded boundary implementation. Finally, we propose a discrete adjoint-based approach to estimate the spatial and temporal errors in the numerical solutions obtained from IMPICE