Time-dependent Nonlinear Partial Differential Equations Research Articles

Motion Prediction of Underwater Sensors

In this work, we will simulate the motion of a single underwater sensor knowing the current velocity to predict its location and velocity during certain time frame using a numerical approach of non-linear time-dependent partial differential equations and develop numerical computer programming code to solve the equations.
 The underwater sensor are used to collect data for many scientific and practical reasons all the sensor collected data without specifying the sensor location and time will be missing lowers valuable information and by simulating the sensor motion numerically will have many values and impact on the underwater sensor industries as this will lead to less power consumption sensors with smaller size and less network coverage required.
 This paper will study the kinetics of the underwater sensor which will resulted to a set of non-linear time-dependent partial differential equations that can be solved analytically and computer programming simulation is developed to solve the equations and predict the motion of underwater sensor.
 Different scenarios considered in the work such as simulating the result for different sensor’s density and the effect on its final position. Also, the result will include the sensor velocity simulation and comparison with the sea current velocity.
 This work is limited to the motion prediction of single underwater sensor and the result is only for mechanical aspect of the problem, the networks connectivity or coverage is out-of-scope.

Motion Prediction of Underwater Sensors

In this work, we will simulate the motion of a single underwater sensor knowing the current velocity to predict its location and velocity during certain time frame using a numerical approach of non-linear time-dependent partial differential equations and develop numerical computer programming code to solve the equations.
 The underwater sensor are used to collect data for many scientific and practical reasons all the sensor collected data without specifying the sensor location and time will be missing lowers valuable information and by simulating the sensor motion numerically will have many values and impact on the underwater sensor industries as this will lead to less power consumption sensors with smaller size and less network coverage required.
 This paper will study the kinetics of the underwater sensor which will resulted to a set of non-linear time-dependent partial differential equations that can be solved analytically and computer programming simulation is developed to solve the equations and predict the motion of underwater sensor.
 Different scenarios considered in the work such as simulating the result for different sensor’s density and the effect on its final position. Also, the result will include the sensor velocity simulation and comparison with the sea current velocity.
 This work is limited to the motion prediction of single underwater sensor and the result is only for mechanical aspect of the problem, the networks connectivity or coverage is out-of-scope.

Characterization and numerical evaluation of flow and blood damage in a pulsatile left ventricular assist device

Each year, thousands of people lose their lives due to left-sided heart failure. Left ventricular assist device (LVAD) is a mechanical pump utilized to aid the function of the left ventricle of the weakened heart. This study investigates flow dynamics, hemolytic and thrombocytopathic characteristics of LVAD with pulsatile flow. To this end, computational fluid dynamics simulations are utilized to predict the levels of hemolysis, thrombosis susceptibility potential and particle residence time in a quantitative manner, and furthermore, we perform dye ejection analysis to understand the pump’s capability to eject old materials. Thus, a set of time-dependent nonlinear partial differential equations are coupled with each other. Blood flow is acquired by solving continuity and momentum equations and levels of hemolysis are computed by coupling two additional scalar transport equations based on a Eulerian transport approach with the governing equations of fluid flow. Additionally, we assess particle residence time and rate of dye ejection according to a continuum-based model. It is depicted, the value of maximum velocity in the pump is much smaller than the axial ones which causes to reduction in red blood cell damage. Average hemolysis obtained its maximum value at the clearance zone, and hemolysis index at the exit of pump is one order lower compared to continuous ones. It is also observed that for all instances of the cycle the maximum residence time concentrates near the outlet domain representing the no accumulation of particles in interior of the pump and capability of the pump to eject material efficiently.

Numerical simulation of ignition of a typical gel fuel particle, based on organic polymer thickener, in a high-temperature air medium

Using the results of previous experimental research by means of high-speed video recording, a mathematical model of ignition was developed for a typical gel fuel particle, based on an organic polymer thickener, in a high-temperature motionless air medium. The structure of such fuel is a rather dense polymer matrix with fine droplets of combustible liquid in the cells. The advantages of gel fuels over liquid rocket propellants dictate their prospects as an energy resource in aerospace industry. The mathematical model of the process under study was developed using the mathematical tools of continuum mechanics and chemical kinetics. It is represented by a system of time-dependent nonlinear partial differential equations with the corresponding initial and boundary conditions. The developed mathematical model describes the process in terms of a limiting mode: the rate of heat supply from the source to the fuel and combustible gas-vapor mixture is limited. This implies that the typical heating time is much longer than the time of chemical reactions. In this case, the factors limiting the process intensity are heat supply and diffusion. Exothermic reactions can be assumed to develop in equilibrium with due consideration of corresponding conditions in the gas medium around the fuel particle. Satisfactory verification results for the mathematical model and numerical algorithm made is possible to conclude that this approach can be used to reliably predict the ignition characteristics of such kind of gel fuels. Gel particles 0.25–2.00 mm in size were considered. They were heated in an air medium at 750–1473 K. The ignition delay times under such conditions ranged from 0.3 to 10.0 s.

3D numerical investigation of the fluid mechanics in a partially liquefied vitreous humor due to saccadic eye movement

Partial vitreous liquefaction (PVL) is a common physical and biochemical degenerative change in the vitreous body in which the liquid component becomes separated from collagen fiber network and this might form the pocket of liquefaction known as lacuna. The main objective of this research is to investigate how the saccade movements influence flow dynamics of the PVL. A three-dimensional model of the vitreous cavity is subjected to saccadic movement and the numerical simulations for various saccade amplitudes and volume fractions are performed. We consider concentric and eccentric configurations of the PVL with the initial spherical shape inside a spherical cavity. In this paper, a specific 3D numerical solver is developed to capture the interface effects and dynamic characteristics of a two-phase viscoelastic-Newtonian fluid flow by using the OpenFOAM CFD. The code is based on a set of time-dependent non-linear partial differential equations (PDE) such as continuity, momentum and constitutive relation for polymeric stresses tensor. The finite volume method with a modified volume-of-fluid model and dynamic mesh technique are used to solve PDEs. Firstly, the validity of the present numerical model was verified by comparing the obtained results with the analytical solutions which demonstrated remarkable agreement. Then, the time- and space-dependent velocity field, shear stress and normal stress distributions were computed and how the PVL responds to the saccadic motions was discussed.

Linkages of calcium-induced calcium release in a cardiomyocyte simulated by a system of seven coupled partial differential equations

Cardiac arrhythmias affect millions of adults in the U.S. each year. This irregularity in the beating of the heart is often caused by dysregulation of calcium in cardiomyocytes, the cardiac muscle cell. Cardiomyocytes function through the interplay between electrical excitation, calcium signaling, and mechanical contraction, an overall process known as calcium-induced calcium release (CICR). A system of seven coupled nonlinear time-dependent partial differential equations (PDEs), which model physiological variables in a cardiac cell, link the processes of cardiomyocytes. Through parameter studies for each component system at a time, we create a set of values for critical parameters that connect the calcium store in the sarcoplasmic reticulum, the effect of electrical excitation, and mechanical contraction in a physiologically reasonable manner. This paper shows the design process of this set of parameters and then shows the possibility to study the influence of a particular problem parameter using the overall model.

An optimal-order error estimate of the lowest-order ELLAM-MFEM approximation to miscible displacement in three space dimensions

An optimal-order error estimate for the lowest-order ELLAM-MFEM time-stepping procedure for the coupled system of time-dependent nonlinear partial differential equations modeling miscible displacement that moves through the three-dimensional subsurface porous media is proved, under a practically reasonable time-step constraint of Δt=O(h). In the procedure, the Eulerian–Lagrangian localized adjoint method (ELLAM) and the mixed finite element method (MFEM) are used to solve the transport equation and the pressure equation, respectively. This mathematically justifies the numerical advantages of the ELLAM-MFEM solution procedure.

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently visualized in terms of binary trees, yields series expansions analogous to the classical Tensor-Train and Hierarchical Tucker tensor formats. By enforcing dynamic orthogonality conditions at each level of the binary tree, we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition. This allows us to effectively compute the solution to high-dimensional time-dependent nonlinear PDEs on tensor manifolds of constant rank, with no need for rank reduction methods. We also propose new algorithms for dynamic addition and removal of modes within each subspace. Numerical examples are presented and discussed for high-dimensional hyperbolic and parabolic PDEs in bounded domains.

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain.

The element-free Galerkin method based on moving least squares and moving Kriging approximations for solving two-dimensional tumor-induced angiogenesis model

The numerical simulation of the tumor-induced angiogenesis process is an useful tool for the prediction of this mechanism and drug targeting using anti-angiogenesis strategy. In the current paper, we study numerically on the continuous mathematical model of tumor-induced angiogenesis in two-dimensional spaces. The studied model is a system of nonlinear time-dependent partial differential equations, which describes the interactions between endothelial cell, tumor angiogenesis factor and fibronectin. We first derive the global weak form of the model and discretize the time variable via a semi-implicit backward Euler method. To approximate the spatial variables of the studied model, we use a meshless technique, namely element-free Galerkin. Also, the shape functions of moving least square and moving Kriging approximations are used in this method. The main difference between two meshless methods proposed here is that the shape functions of moving least squares approximation do not satisfy Kroncker’s delta property, while moving Kriging technique satisfies this property. Also, both techniques do not require the generation of a mesh for approximation, but a background mesh is needed to compute the numerical integrations, which are appeared in the derived global weak form. The full-discrete scheme obtained here gives the linear system of algebraic equations that is solved via an iterative method, namely biconjugate gradient stabilized with zero-fill incomplete lower upper (ILU) preconditioner. Some numerical simulations are provided to illustrate the ability of the presented numerical methods, which show the endothelial cell migration in response to the tumor angiogenesis factors during angiogenesis process as well.

Conservative or dissipative quasi-interpolation method for evolutionary partial differential equations

Based on quasi-interpolation, we propose a new meshless conservative or dissipative method for nonlinear time-dependent partial differential equations. Using the method of lines, we first discretize the equation in space with the quasi-interpolation method, then employ the average vector field method in time discretization to derive the final numerical scheme. The method not only inherits the conservation or dissipation property of the equation but also has the meshless feature since we use the nonuniform grids in spatial discretization. Several numerical examples are presented to demonstrate the effectiveness of the proposed method.

A Posteriori Error Estimation and Adaptivity for Nonlinear Parabolic Equations using IMEX-Galerkin Discretization of Primal and Dual Equations

While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remain challenging...

On multilevel RBF collocation to solve nonlinear PDEs arising from endogenous stochastic volatility models

The main aim of this paper is the analytical and numerical study of a time-dependent second-order nonlinear partial differential equation (PDE) arising from the endogenous stochastic volatility model, introduced in [Bensoussan, A., Crouhy, M. and Galai, D., Stochastic equity volatility related to the leverage effect (I): equity volatility behavior. Applied Mathematical Finance, 1, 63–85, 1994]. As the first step, we derive a consistent set of initial and boundary conditions to complement the PDE, when the firm is financed by equity and debt. In the sequel, we propose a Newton-based iteration scheme for nonlinear parabolic PDEs which is an extension of a method for solving elliptic partial differential equations introduced in [Fasshauer, G. E., Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43, 423–438, 2002]. The scheme is based on multilevel collocation using radial basis functions (RBFs) to solve the resulting locally linearized elliptic PDEs obtained at each level of the Newton iteration. We show the effectiveness of the resulting framework by solving a prototypical example from the field and compare the results with those obtained from three different techniques: (1) a finite difference discretization; (2) a naive RBF collocation and (3) a benchmark approximation, introduced for the first time in this paper. The numerical results confirm the robustness, higher convergence rate and good stability properties of the proposed scheme compared to other alternatives. We also comment on some possible research directions in this field.

Adaptive multilevel trust-region methods for time-dependent PDE-constrained optimization

We present a class of adaptive multilevel trust-region methods for the efficient solution of optimization problems governed by time-dependent nonlinear partial differential equations with control constraints. The algorithm is based on the ideas of the adaptive multilevel inexact SQP-method from \[26], \[27]. It is in particular well suited for problems with time-dependent PDE constraints. Instead of the quasi-normal step in a classical SQP method which results in solving the linearized PDE sufficiently well, in this algorithm a (nonlinear) solver is applied to the current discretization of the PDE. Moreover, different discretizations and solvers for the PDE and the adjoint PDE may be applied. The resulting inexactness of the reduced gradient in the current discretization is controlled within the algorithm. Thus, highly efficient PDE solvers can be coupled with the proposed optimization framework. The algorithm starts with a coarse discretization of the underlying optimization problem and provides during the optimization process implementable criteria for an adaptive refinement strategy of the current discretization based on error estimators. We prove global convergence to a stationary point of the infinite-dimensional problem. Moreover, we illustrate how the adaptive refinement strategy of the algorithm can be implemented by using a posteriori error estimators for the state and the adjoint equation. Numerical results for a semilinear parabolic PDE-constrained problem with pointwise control constraints are presented.

Nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon-nanotube reinforced composite beams

This study aims at numerically analyzing the nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams with various end conditions subjected to a harmonic transverse load. Introducing a generalized displacement field including various beam theories, employing Hamilton’s principle and taking into account geometrical nonlinearity and initial imperfection, three nonlinear coupled equations and associated boundary expressions are obtained for geometrically imperfect FG-CNTRC beams. These equations formulate the longitudinal, transverse and rotational motions of FG-CNTRC beams. An efficient multistep numerical solution approach based on the generalized differential quadrature (GDQ) method, a numerical Galerkin-based scheme and time periodic discretization is employed to convert the time-dependent nonlinear partial differential equations (PDEs) into a Duffing-type nonlinear set of ordinary differential equations (ODEs) which can be solved via the pseudo arc-length continuation technique. Nonlinear resonant dynamics characteristics are illustrated in the form of frequency-response and force-response curves; highlighting the influences of initial geometrical imperfection, geometrical parameters, excitation frequency and boundary conditions.

On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

In this article, one-step semi-Lagrangian method is investigated for computing the numerical solutions of the first-order time-dependent nonlinear partial differential equations in 1D with initial and boundary conditions. This method is based on Lagrangian trajectory or the integration from the departure points to the arrival points (regular nodes) and Runge–Kutta method for ordinary differential equations. The departure points are traced back from the arrival points along the trajectory of the path. The convergence and stability are studied for the implicit and explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations.

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

A novel parameterized non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and efficient approach for model reduction of parameterized partial differential equations (P-PDEs). Over existing parameterized reduced order models (P-ROM) (most of them are based on the reduced basis method), it is non-intrusive and independent on partial differential equations and computational codes. During the training process, the Smolyak sparse grid method is used to select a set of parameters over a specific parameterized space (Ωp∈RP). For each selected parameter, the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function sets for any parameters over Ωp can be obtained using an interpolation method. The P-NIROM can then be constructed by using our recently developed technique (Xiao et al., 2015 [41,42]) where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the reduced space.The new P-NIROM technique has been applied to parameterized Navier–Stokes equations and implemented with an unstructured mesh finite element model. The capability of this P-NIROM has been illustrated numerically by two test cases: flow past a cylinder and lock exchange case. The prediction capabilities of the P-NIROM have been evaluated by varying the viscosity, initial and boundary conditions. The results show that this P-NIROM has captured the quasi-totality of the details of the flow with CPU speedup of three orders of magnitude. An error analysis for the P-NIROM has been carried out.

OpenMP Fortran and C programs for solving the time-dependent Gross–Pitaevskii equation in an anisotropic trap

We present new version of previously published Fortran and C programs for solving the Gross-Pitaevskii equation for a Bose-Einstein condensate with contact interaction in one, two and three spatial dimensions in imaginary and real time, yielding both stationary and non-stationary solutions. To reduce the execution time on multicore processors, new versions of parallelized programs are developed using Open Multi-Processing (OpenMP) interface. The input in the previous versions of programs was the mathematical quantity nonlinearity for dimensionless form of Gross-Pitaevskii equation, whereas in the present programs the inputs are quantities of experimental interest, such as, number of atoms, scattering length, oscillator length for the trap, etc. New output files for some integrated one- and two-dimensional densities of experimental interest are given. We also present speedup test results for the new programs.

Soliton Kernels for Solving PDEs

There are many important applications in the fields of computer experiments, response surface modeling, finance and image processing, where some special types of nonstandard kernels performed better than standard kernels. These kernels are more appropriate than standard kernels when looking at special solutions of partial differential equations (PDEs). For example some nonlinear time-dependent PDEs have soliton like solutions, so soliton kernels would more suit to approximate the solution. In this work, we recover the solution of equal width equation using soliton kernels.

A bounded linear integrator for some diffusive nonlinear time-dependent partial differential equations

We propose a numerical method to approximate the solutions of generalized forms of two bi-dimensional models of mathematical physics, namely, the Burgers–Fisher and the Burgers–Huxley equations. In one-dimensional form, the literature in the area gives account of the existence of analytical solutions for both models, in the form of traveling-wave fronts bounded within an interval I of the real numbers. Motivated by this fact, we propose a finite-difference methodology that guarantees that, under certain analytical conditions on the model and computer parameters, estimates within I will evolve discretely into new estimates which are likewise bounded within I. Additionally, we establish the preservation in the discrete domain of the skew-symmetry of the solutions of the models under study. Our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically. Our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations.