Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well.

A Modeling Study of the Productivity of Horizontal Wells in Hydrocarbon-Bearing Reservoirs: Effects of Fracturing Interference

Commercial production from hydrocarbon-bearing reservoirs with low permeability usually requires the use of horizontal well and hydraulic fracturing for the improvement of the fluid diffusivity in the matrix. The hydraulic fracturing process involves the injection of viscous fluid for fracture initiation and propagation, which alters the poroelastic behaviors in the formation and causes fracturing interference. Previous modeling studies usually focused on the effect of fracturing interference on the multicluster fracture geometry, while the related productivity of horizontal wells is not well studied. This study presents a modeling workflow that utilizes abundant field data including petrophysical, geomechanical, and hydraulic fracturing data. It is used for the quantification of fracturing interference and its correlation with horizontal well productivity. It involves finite element and finite difference methods in the numeralization of the fracture propagation mechanism and porous media flow problems. Planar multistage fractures and their resultant horizontal productivity are quantified through the modeling workflow. Results show that the smaller numbers of clusters per stage, closer stage spacings, and lower fracturing fluid injection rates facilitate even growth of fractures in clusters and stages and reduce fracturing interference. Fracturing modeling results are generally correlated with productivity modeling results, while scenarios with stronger fracturing interference and greater stimulation volume/area can still yield better productivity. This study establishes the quantitative correlation between fracturing interference and horizontal well productivity. It provides insights into the prediction of horizontal well productivity based on fracturing design parameters.

Mathematical models for blood flow in elastic vessels: Theory and numerical analysis

<p>In this thesis we study model equations that describe the propagation of pulsatile flow in elastic vessels. Since dealing with the Navier-Stokes equations is a very difficult task, we derive new asymptotic weakly non-linear and weakly-dispersive Boussinesq systems. Properties of the these systems, such as the well-posedness, and existence of travelling waves are being explored. Finally, we discretize some of the new model equations using finite difference methods and we demonstrate their applicability to blood flow problems. First we introduce the basic equations that describe f luid flow in elastic vessels and previously derived systems. We also review previously derived model equations for fluid flow in elastic tubes. We start with the description of the equations of motion of elastic vessel. Then wederive asymptotically Boussinesq systems for fluid flow in elastic vessels. Because these systems are weakly non-linear and weakly dispersive we expect then to have solitary waves as special solutions. We explore some possibilities by construction analytical solutions. After that we continue the derivation of the previous chapter. We derive a general system where the horizontal velocity is evaluated at any distance from the center of the tube. Special emphasis is paid on the case of constant radius vessels. We also derive unidirectional models and obtain the dissipative Boussinesq system by taking the viscosity effects into account. There is also an alternative derivation of the general system when considering the equations of potential flow. We show that the two different derivations lead to the same system. The alternative derivation is based on asymptotic series expansions. Then we develop finite difference methods for the numerical solution of the BBM equation and for the classical Boussinesq system studied in the previous chapters. Finally, we demonstrate the application of the new models to blood flow problems. By performing several numerical simulations.</p>

Mathematical models for blood flow in elastic vessels: Theory and numerical analysis

<p>In this thesis we study model equations that describe the propagation of pulsatile flow in elastic vessels. Since dealing with the Navier-Stokes equations is a very difficult task, we derive new asymptotic weakly non-linear and weakly-dispersive Boussinesq systems. Properties of the these systems, such as the well-posedness, and existence of travelling waves are being explored. Finally, we discretize some of the new model equations using finite difference methods and we demonstrate their applicability to blood flow problems. First we introduce the basic equations that describe f luid flow in elastic vessels and previously derived systems. We also review previously derived model equations for fluid flow in elastic tubes. We start with the description of the equations of motion of elastic vessel. Then wederive asymptotically Boussinesq systems for fluid flow in elastic vessels. Because these systems are weakly non-linear and weakly dispersive we expect then to have solitary waves as special solutions. We explore some possibilities by construction analytical solutions. After that we continue the derivation of the previous chapter. We derive a general system where the horizontal velocity is evaluated at any distance from the center of the tube. Special emphasis is paid on the case of constant radius vessels. We also derive unidirectional models and obtain the dissipative Boussinesq system by taking the viscosity effects into account. There is also an alternative derivation of the general system when considering the equations of potential flow. We show that the two different derivations lead to the same system. The alternative derivation is based on asymptotic series expansions. Then we develop finite difference methods for the numerical solution of the BBM equation and for the classical Boussinesq system studied in the previous chapters. Finally, we demonstrate the application of the new models to blood flow problems. By performing several numerical simulations.</p>

Quantum-mechanical model of dielectric losses in nanometer layers of solid dielectrics with hydrogen bonds at ultra-low temperatures

Upon based the finite difference methods construct the solutions for Liouville quantum kinetic equation linearized by the external field, in complex with the stationary Schrodinger equation and the Poisson operator equation, for an ensemble of non-interacting hydrogen ions (protons) migrating in the field of a crystal lattice perturbed by a variable polarizing field. The influence of the phonon subsystem is not taken into account. The equilibrium (non-balanced) proton density matrix is calculated using quantum Boltzmann statistics. The temperature spectra of dielectric losses tangent angle for hydrogen bonded crystals (HBC) in a wide temperature range (50–550 K) are calculated. At the theoretical level detected the effects of nano-crystalline states (1–10 nm) during the polarization of HBC in the region of ultra-low temperatures (4–25 K).

On exponential fitting of finite difference methods for heat equations

In this article, a new kind of finite difference scheme that is exponentially fitted, inspired from Fourier analysis, for a fourth space derivative was developed for solving diffusion problems. Dispersion relation and local truncation error of the method were discussed. Stability analysis of the method revealed that it is conditionally stable. Compared to the corresponding fourth order classical scheme in the literature, the proposed scheme is efficient and accurate. Mathematics Subject Classification (2020): 65M06, 65N06. Keywords: Exponential fitting, Finite difference, Local truncation error, Heat equations.

Some Finite Difference Methods to Model Biofilm Growth and Decay: Classical and Non-Standard

The study of biofilm formation is undoubtedly important due to micro-organisms forming a protected mode from the host defense mechanism, which may result in alteration in the host gene transcription and growth rate. A mathematical model of the nonlinear advection–diffusion–reaction equation has been studied for biofilm formation. In this paper, we present two novel non-standard finite difference schemes to obtain an approximate solution to the mathematical model of biofilm formation. One explicit non-standard finite difference scheme is proposed for biomass density equation and one property-conserving scheme for a coupled substrate–biomass system of equations. The nonlinear term in the mathematical model has been handled efficiently. The proposed schemes maintain dynamical consistency (positivity, boundedness, merging of colonies, biofilm annihilation), which is revealed through experimental observation. In order to verify the accuracy and effectiveness of our proposed schemes, we compare our results with those obtained from standard finite difference schemes and earlier known results in the literature. The proposed schemes (NSFD1 and NSFD2) show good performance. The NSFD2 scheme reveals that the processes of biofilm formation and nutritive substrate growth are intricately linked.

High-precision quantum algorithms for partial differential equations

Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity poly(1/ϵ), where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be poly(d,log⁡(1/ϵ)), where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

Professional competence development when teaching computational informatics.

Bachelors and graduate students are offered in the course of teaching computational informatics, the ability to solve non-standard mathematical problems, which, as a rule, are not included in the content of teaching computational informatics. The article aimed to analyze the application effectiveness of non-standard mathematical problems in the course of teaching computational informatics, elaboration of constructive computational solution algorithms of inverse problems for differential equations, during which the bachelors and graduate students develop own professional competencies. The research conducted a review of previous literature on the topic. Formulation of the inverse problem for differential equations for the investigation of which the computational mathematics finite difference methods are applied, is presented. In the course of investigation, it was revealed that at elaborating the constructive computational algorithms of its solution, the bachelors and graduate students develop not only fundamental knowledge in the field of applied and computational mathematics, computational informatics methods, but also develop the professional competences, including computational thinking. Key words: professional competence; computational informatics; computational mathematics methods; non-standard.