Nonlinear Initial-boundary Value Problem Research Articles

An explicit Jacobian for Newton's method applied to nonlinear initial boundary value problems in summation-by-parts form

<p>We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.</p>

Encapsulated generalized summation-by-parts formulations for curvilinear and non-conforming meshes

We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development, energy stable discretizations of nonlinear and variable coefficient initial boundary value problems can be formulated in simple and straightforward ways using high-order accurate operators of generalized summation-by-parts type. Encapsulated features on a single computational block or element may include polynomial bases, tensor products as well as curvilinear coordinate transformations. Moreover, through the use of inner product preserving interpolation or projection, the global summation-by-parts property is extended to arbitrary multi-block or multi-element meshes with non-conforming nodal interfaces.

Nonlinear boundary conditions for initial boundary value problems with applications in computational fluid dynamics

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in skew-symmetric form, including dissipative terms. The complete procedure has two main ingredients. Firstly, the energy rate in terms of a surface integral with boundary terms is derived. Secondly, we bound the surface integral by deriving new nonlinear boundary procedures for boundary conditions with non-zero data. The new nonlinear boundary procedure generalises the well known characteristic boundary procedure for linear problems to the nonlinear setting.To introduce the procedure, a skew-symmetric scalar IBVP encompassing the linear advection equation and Burgers equation is analysed. Once the continuous analysis is done, we show that energy stable nonlinear discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions. The scalar analysis is subsequently repeated for general nonlinear systems of equations. Finally, the new boundary procedure is applied to four important IBVPs in computational fluid dynamics: the incompressible Euler and Navier-Stokes, the shallow water and the compressible Euler equations.

Modeling of Dynamic Thermo-Elastic-Viscous-Plastic Deformation of Flexible Shallow Reinforced Shells

A mathematical model of non-isothermal elastic-viscous-plastic deformation of flexible shallow shells with multidirectional reinforcement structures has been developed. Wave processes and weak resistance to transverse shear in curved panels are modeled in terms of Ambartsumian’s bending theory. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The composition components are assumed to be isotropic materials, and their plasticity is described by the flow theory with a loading function depending on the strain rate and temperature. The connectedness of the thermomechanical problem under dynamic loading of composite shallow shells is taken into account. In the transverse direction of constructions, the temperature is approximated by a 7th order polynomial. The formulated two-dimensional nonlinear initial-boundary value problem is solved using an explicit numerical scheme of time steps. The thermo-elastic-visco-plastic and thermo-elastic-plastic behavior of fiberglass and metal-composite shallow shells orthogonally reinforced in two tangential directions, loaded in the transverse direction by an air blast wave, has been studied. It is shown that flexible curved fiberglass panels at certain points can additionally heat up by 14…27°C, and similar metal-composite conctructions – by 70°С or more. In this case, peak temperature values are kept at short-term intervals – on the order of fractions of 1 ms. It is shown that, unlike flexible plates, similar shallow shells (with the same reinforcement structure and the same characteristic dimensions) under dynamic loading in the transverse direction must be calculated not only taking into account the dependence of the plastic properties of the composition components on their strain rate, but also taking into account thermal response in such thin-walled constructions. A more intense inelastic deformation of curved composite panels is observed when they are loaded from the side of the convex front surface.

Critical exponent for global existence of solutions to the Schrödinger equation with a nonlinear boundary condition

We consider the nonlinear initial boundary value problem of one dimensional Schrödinger equation with the power nonlinearity of order q in boundary conditions. Our purpose in this paper is to prove that if q>2 and the initial data are small in suitable norms, then we have a global in time of solutions in a scale invariant space. Furthermore, we present a blow up result of small amplitude solutions when 1<q≤2 and derive an upper-bound of existence time Our results indicate that the quadratic nonlinearity is a critical exponent.

ТЕМПЕРАТУРНОЕ ПОЛЕ ОДНОРОДНОЙ КВАДРАТНОЙ ОБЛАСТИ С ДВИЖУЩИМИСЯ БЕЗ УСКОРЕНИЯ СМЕЖНЫМИ СТОРОНАМИ ПРИ ГРАНИЧНЫХ УСЛОВИЯХ ПЕРВОГО РОДА

A square area with homogeneous thermal and physical characteristics, deformed preserving 2-D similarity, is investigated. At the initial moment of time, two adjacent sides start moving respectively towards the abscissa and ordinate axes with constant speed while remaining equidistant to the other two adjacent sides (the fixed and moving sides are kept at different constant temperatures). A nonlinear initial boundary value problem with boundary conditions of the first kind and special coordinates immobilizes the moving boundary of the area into a fixed one with the corresponding transformation of the initial boundary value problem for the fixed boundaries with respect to the multiplicative variable of two unknown functions, which are defined by additional initial boundary values. These were solved by the successive application of integral sine transformations on pseudo-space variables. This enables the solution of the original problem to be notated analytically using special quadratures. The computational experiment proved the correctness of the solution and the absolute fulfillment of the initial conditions. The results also illustrate the adequacy of the qualitative calculations for the heating process of a quadratic area with moving adjacent boundaries. This approach can be applied to the differently directed motion of adjacent boundaries, to uniformly retarded or uniformly accelerated motion. Considering that Fourier's and Fick's laws are mathematically similar, the solution and its generalization are of practical importance in describing mass transfer processes, such as crystallization or dissolution.

A class of linearization-based collocation methods for initial value and boundary value engineering problems

Nonlinear initial value and boundary value problems are generally classified as two different categories of problems and the methods utilized to solve them are also different. This paper proposed a uniform linearization-collocation framework through which the initial value and boundary value problems can be solved in a similar way. An optional pre-conditioning procedure is provided to guarantee the computational accuracy. Rigorous analysis about the convergence rate is given. Different from the commonly seen convergence rate analysis based on one-dimensional system and mean-value theorem, this analysis is based on multidimensional system and polynomial approximation. A perturbed orbit propagation problem, a radiation heat transfer problem and an orbit optimization problem are solved to verify the validity and efficiency of the proposed methods. The proposed methods not only outperform the highly optimized MATLAB built-in integrator ODE45 in orbit propagation, but also outperform the bvp4c and bvp5c in solving the boundary value problem. Most importantly, the proposed methods are capable of solving some problems where the bvp4c and bvp5c are invalid. The quasi-linearization-collocation method proved to be a highly accurate, efficient and stable solver for ordinary differential equations with various types of constraints.

Determination of thermophysical characteristics in a nonlinear inverse heat transfer problem

This paper presents a detailed and developed method for finding soil nonlinear thermophysical characteristics. Two-layer container complexes are constructed. The side faces of which are thermally insulated, so that one can use the one dimensional heat equation. In order not to solve the boundary value problem with a contact discontinuity and not to lose the accuracy of the solution method, a temperature sensor was placed at the junction of the two media, and a mixed boundary value problem is solved in each region of the container. In order to provide the initial data for the inverse coefficient problem, two temperature sensors are used: one sensor was placed on the open border of the container and recorded the temperature of the soil at this border, and the second sensor was placed at a short distance from the border, which recorded the air temperature. The measurements were carried out in the time interval (0,4tmax). At first, the initial-boundary value problem of nonlinear thermal conductivity equation with temperature dependent coefficients of thermal conductivity, heat capacity, heat transfer and material density is investigated numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas’ method, and the nonlinear difference problem is solved by the Newton’s method. Assuming the constancy of all thermophysical parameters, except for the thermal conductivity of the material, on the segment (0,tmax), the quadratic convergence of Newton’s method is proved and the thermal conductivity is calculated, while the solution of the linearized difference problem was taken as the initial approximation for the Newton’s method. By freezing all other thermophysical parameters, and using the measured temperatures in the segment (tmax,2tmax), the coefficient of specific heat of the material is found. Proceeding in a similar way, using the initial data on the segment (2tmax,3tmax), the density of the soil is determined. Also, based on the measured initial data on the segment (3tmax,4tmax), the heat transfer coefficient is determined.Based on the experimentally measured data, at each time interval, the corresponding functional is minimized using the gradient descent method. The differentiation of a nonlinear difference problem with respect to the desired parameter method is used to find the gradient of the functional. This method allows you to find the functional gradient and the damping coefficient of the gradient descent in an explicit form. In this case, the compilation and solution of the conjugate problem, which is a frequently used method for solving the inverse problems, is not needed.The performed numerical calculations show that for small time intervals the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and black soil. Also the dependence of the temperature difference at the boundary between the environment and the soil for the heat transfer coefficient was shown.

CMMSE: Analysis of order reduction when Lawson methods integrate nonlinear initial boundary value problems

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction‐diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper.

Consistent discrete global dynamics of a general initial boundary value problem for hepatitis B virus infection with capsids and adaptive immunity

In this paper, we employ the nonstandard finite difference method to discretize a nonlinear initial boundary value problem that describes the global dynamics of hepatitis B virus infection with adaptive immunity. The model considered incorporates intracellular HBV DNA-containing capsids, a general incidence function and two categories of infected hepatocytes: exposed infected hepatocytes and productively infected hepatocytes. Adaptive immunity cells are recruited to attack all infected hepatocytes and free virus particles. The dynamics of fundamental properties of both discrete and continuous models are thoroughly studied, and it is shown that the discrete system is dynamically consistent with the continuous model. These properties include existence, nonnegativity and boundedness of solutions as well as the global stability of spatially homogeneous equilibria. We carry out some numerical simulations to support the theoretical findings and illustrate the behaviour of the solution in space and time.

EXPERIMENTAL DATA AND THE NONLINEAR INVERSE PROBLEM OF HEAT TRANSFER

In the paper the development a method for finding the nonlinear heat-conducting characteristics of the soil is being presented. Two-layer container complexes have been created, the side faces of which are thermally insulated so that the 1D heat equation can be used. In order not to solve the boundary value problem with a contact discontinuity and lose the accuracy of the method’s solution, a temperature sensor was placed at the junction of two media, and a mixed boundary value problem is solved in each area (container). To provide the initial data with an inverse coefficient problem, two temperature sensors are used: one sensor was placed at the open boundary of the container and recorded the soil temperature at this boundary, and the second sensor was placed at a short distance from the boundary, which recorded the air temperature. The measurements were carried out on the time interval &nbsp;First, the initial-boundary value problem of thermal conductivity with nonlinear coefficients of thermal conductivity, heat capacity, heat transfer, and material density are studied numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Based on the measured data of the complex, special functionals are constructed and the thermal conductivity coefficient , density , specific heat capacity , heat transfer coefficient are found, which depend on the temperature of the material. Based on the experimentally measured data, the corresponding functional is minimized on each time interval using the gradient descent method. All thermophysical characteristics for a container with clay were found with a relative error of 5%. &nbsp;

AN INITIAL-BOUNDARY VALUE PROBLEM FOR KELVIN-VOIGT EQUATIONS WITH (p(x), q(x),m(x))- STRUCTURE

The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many cases,&nbsp; is not easy, even in some cases it is impossible. However, by showing some qualitative properties of its solution, one can find answers to such questions. For example, by establishing the blow up in a finite time property of a solution, one can show that a solution does not exist globally in time. Such way, in last years, the investigating the quality properties of solutions as localization and/or blow up in a finite time, has been developing rapidly In this paper, we study the initial-boundary value problem for the Kelvin-Voigt equations with both of diffusion and relaxation term, modified by (p(x),q(x))-Laplacian, respectively, and with the variable exponent damping term. The damping term in the equation realizes as nonlinear source term. In this work, we study the nonlinear initial-boundary value problem for generalized Kelvin-Voigt equations describing the motion of incompressible viscoelastic non-Newtonian fluids. The equations generalized by replacing the diffusion and relaxation terms in equation with p(x)-Laplacian and q(x)-Laplacian, respectively, and adding a nonlinear absorption term with variable exponents and coefficients. The definition of a weak solution is given. Under suitable conditions for variable exponents and coefficients, and data of the problem, the blow up of weak solutions are established.

Solution of a nonlinear heat transfer problem based on experimental data

The paper develops a method for solving nonlinear equations of heat conduction. Two-layer container complexes have been created, the side faces of which are thermally insulated so that the 1D heat equation can be used. In order not to solve the boundary value problem with a contact discontinuity and lose the accuracy of the solution method, a temperature sensor was placed at the junction of two media, and a mixed boundary value problem is solved in each area (container). To provide initial data for the initial boundary value problem, three temperature sensors were used: two sensors measure air temperatures at the left and right boundaries of the container complex; the third sensor measures the temperature of the soil at the junction of two media. The paper numerically investigates the initial-boundary problem of heat conduction with nonlinear coefficients of heat conduction, heat capacity, heat transfer and density of the material. To solve a nonlinear initial-boundary value problem, the grid method is used. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar sweep method, and the nonlinear difference problem is solved by the Newton method. On the basis of an a priori estimate of the solution of a nonlinear difference problem, we prove the convergence of the second degree of Newton’s method. The numerical calculations carried out show that, for small time intervals, the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%

On a Nonlinear Initial—Boundary Value Problem with Venttsel Type Boundary Conditions Arizing in Homogenization of Complex Heat Transfer Problems

We consider a non-standard nonlinear singularly perturbed 2D initial-boundary value problem with Venttsel type boundary conditions, arising in homogenization of radiative-conductive heat transfer problems. We establish existence, uniqueness and regularity of a weak solution v. We obtained estimates for the derivatives Dtv, D12v, D22v, D1D2v with a qualified order in the small parameter ε.

An existence result for a strongly nonlinear parabolic equations with variable nonlinearity

We prove the existence of a solution for the strongly nonlinear parabolic initial boundary value problem associated to the equation ut − div a(x, t, ∇u) + g(x, t, u, ∇u) = f, where the vector field a(x, t, ξ) exhibits non-standard growth conditions.

Nonlinear and linearised primal and dual initial boundary value problems: When are they bounded? How are they connected?

Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an energy bound but the linearised version does not (or vice versa). In this paper we attempt to resolve that contradiction and relate nonlinear energy conserving and bounded initial boundary value problems to their linearised versions and the related dual problems. We start by showing that a specific skew-symmetric form of the primal nonlinear problem leads to energy conservation and a bound. Next, we show that this specific form together with a non-standard linearisation procedure preserves these properties for the new slightly modified linearised problem. We proceed to show that the corresponding linear and nonlinear dual (or self-adjoint) problems also have bounds and conserve energy due to this specific formulation. Next, the implication of the new formulation on the choice of boundary conditions is discussed. A straightforward nonlinear and linear analysis may lead to a different number and type of boundary conditions required for an energy bound. We show that the new formulation sheds some light on this contradiction. We conclude by illustrating that the new continuous formulation automatically leads to energy stable and energy conserving numerical approximations for both linear and nonlinear primal and dual problems if the approximations are formulated on summation-by-parts form.

THREE-DIMENSIONAL NONSTATIONARY FLOW OF TWO IMMISCIBLE FLUIDS IN A ROTATING HORIZONTAL CYLINDER

The three-dimensional nonstationary axisymmetric flow of two immiscible viscous fluids in a rotating horizontal cylinder is studied. The interface between the fluids is assumed to be flat and undeformable. The full energy condition at the interface is taken into account. The mass forces are neglected. The velocity field of the Himenz type is considered. For the solution of the nonlinear initial boundary value problem, the modified Galerkin method and the Runge-Kutta method have been applied. It is shown that, with an increase of time, the numerical solution of the nonstationary problem tends to the stationary regime.

The interior-boundary Strichartz estimate for the Schrödinger equation on the half-line revisited

In recent papers, it was shown for the biharmonic Schrödinger equation and 2D Schrödinger equation that Fokas method-based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interior-boundary Strichartz estimates for the Schrödinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low regularity solutions for the nonlinear Schrödinger equation (NLS) with Neumann boundary condition and a coupled system of NLS equations defined on the half line with Dirichlet/Neumann boundary conditions.

Application of reaction-diffusion equations for modeling human and breeding site attraction movement behavior of Aedes aegypti mosquito.

This paper shows how biological population dynamic models in the form of coupled reaction-diffusion equations with nonlinear reaction terms can be applied to heterogeneous landscapes. The presented systems of coupled partial differential equations (PDEs) combine the dispersal of disease-vector mosquitoes and the spread of the disease in a human population. Realistic biological dispersal behavior is taken into account by applying chemotaxis terms for the attraction to the human host and the attraction of suitable breeding sites. These terms are capable of generating the complex active movement patterns of mosquitoes along the gradients of the attractants. The nonlinear initial boundary value problems are solved numerically for geometries of heterogeneous landscapes, which have been imported from geographic information system data to construct a general-purpose finite-element solver for systems of coupled PDEs. The method is applied to the dispersal of the dengue disease vector for Aedes aegypti in a small-scale rural setting consisting of small houses and different breeding sites, and to a large-scale section of the suburban zone of a metropolitan area in Vietnam. Numerical simulations illustrate how the setup of model equations and geographic information can be used for the assessment of control measures, including the spraying patterns of pesticides and biological control by inducing male sterility.

Nonlinear systems of parabolic IBVP: A stable super-supraconvergent fully discrete piecewise linear FEM

The main purpose of this paper is the design of discretizations for second order nonlinear parabolic initial boundary value problems which are stable and present second order of convergence with respect to H1-discrete norms. The convergence results are established assuming that the solutions are in H3. The stability analysis of numerical methods around a numerical solution requires the uniform boundness of such solution. Although such bounds are usually taken as an assumption, in this paper these will be deduced as a corollary of suitable error estimates. As the methods can be simultaneously seen as piecewise linear finite element methods and finite difference methods, the convergence results can be seen simultaneously as superconvergence results and supraconvergence results. Numerical results illustrating the sharpness of the smoothness assumptions and an application to simulation of the solar magnetic field in the umbra (the central zone of a sunspot) are also included.