Time-dependent Partial Differential Equations Research Articles

Parameter Robust Preconditioning by Congruence for Multiple-Network Poroelasticity

The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multiporosity/multipermeability systems. These equations generalize Biot's equations, which describe the mechanics of the one network case. The efficient numerical solution of the MPET equations is challenging, in part due to the complexity of the system and in part due to the presence of interacting parameter regimes. In this paper, we present a new strategy for efficiently and robustly solving the MPET equations numerically. In particular, we discuss an approach to formulating finite element methods and associated preconditioners for the MPET equations based on simultaneous diagonalization of the element matrices. We demonstrate the technique for the multicompartment Darcy equations, with large exchange variability, and the MPET equations for a nearly incompressible medium with large exchange variability. The approach is based on designing transformations of variables that simultaneously diagonalize (by congruence) the equations' key operators and subsequently constructing parameter-robust block diagonal preconditioners for the transformed system. The proposed approach is supported by theoretical considerations as well as by numerical results.

Role of exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow: a Riemann-sum approximation approach

Our study probes the impact of an exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow in a curved concentric cylinder. A two-step method of solution has been employed in the treatment of the governing momentum equation. Accordingly, the exact solution of the time-dependent partial differential equation is derived in terms of the Laplace variable. The Laplace domain solution is then transformed to the time domain using a numerical inversing scheme known as Riemann-sum approximation. The effect of the various dimensionless parameters involved in the problem on the Dean velocity, skin drags and Dean vortex are illustrated graphically. It was established that maximum Dean velocity is due to an exponentially growing time-dependent pressure gradient. However, the instability of the Dean vortex is rendered less effective by reducing time and applying an exponentially decaying time-dependent pressure gradient.

Reverse-Time Migration by Combining Laplacian Filtering With Wavefield Decomposition for High-Resolution Subsurface Imaging

Reverse-time migration (RTM) is considered to be one of the most accurate migration methods, especially for imaging geologically complex structures. However, the conventional RTM using cross-correlation imaging condition is subject to strong migration artifacts. Recent researches on electromagnetic wavefield decomposition exhibit that the method can greatly suppresses the internal reflection noise, however, it is still subject to residual noise. To further eliminate the low-frequency noise to obtain high-resolution subsurface image, a novel electromagnetic imaging scheme by combining a Laplacian filtering with wavefield decomposition is presented, and the pseudospectral time-domain (PSTD) method is introduced to solve the time-dependent partial differential equations efficiently. In this work, we apply the Laplacian filtering on each wavefield snapshot in space domain to sharpen the field distribution. The filtered wavefields are further decomposed into downgoing and upgoing components for final imaging. Numerical experiments show that the proposed method balances the amplitudes of reflectors, and exhibits higher resolution compared to the conventional RTM.

Structure-Preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods

Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural solution properties, which we enforce as additional constraints on the solution. We consider in particular the class of convex constraints, which includes positivity and monotonicity. By embedding such constraints as a postprocessing convex optimization procedure, we can compute solutions that satisfy general types of convex constraints. For certain types of constraints (including positivity and monotonicity), the optimization is a filter, i.e., a norm-decreasing operation. We provide a variety of tests on one-dimensional time-dependent PDEs that demonstrate the method's efficacy, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.

Numerical Solutions of Time Dependent Partial Differential Equations via HAAR Wavelets

An effective wavelet based scheme coupled with finite difference is used for the solution of two nonlinear time dependent problems namely: Burgers' and Boussinesq equations. These equations have wide-spread application in many fields such as viscous medium, turbulence , uid dynamics, infiltration phenomena etc. The proposed scheme convert the partial differential equations (PDE) to system of algebraic equations. The obtained system can be solved easily. In this paper convergence of the scheme is also discussed to show validity of the technique. Effectiveness of the scheme is shown with the help of test problems. Numerical results verify that the suggested scheme is more accurate, convenient, fast and require low computational cost.

Uniqueness for time-dependent inverse problems with single dynamical data

In this work, we investigate the shape identification and coefficient determination associated with two time-dependent partial differential equations in two dimensions. We consider the inverse problems of determining a convex polygonal obstacle and the coefficient appearing in the wave and Schrödinger equations from a single dynamical data. With the near-field data, we first prove that the sound speed of the wave equation together with its contrast support of convex-polygon type can be uniquely determined, then establish a uniqueness result for recovering an electric potential as well as its support appearing in the Schrödinger equation. As a consequence of these results, we demonstrate a uniqueness result for recovering the refractive index of a medium from a single far field pattern at a fixed frequency in the time-harmonic regime.

Numerical simulation of a prostate tumor growth model by the RBF-FD scheme and a semi-implicit time discretization

The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains.

A DPG-based time-marching scheme for linear hyperbolic problems

The Discontinuous Petrov–Galerkin (DPG) method is a widely employed discretization method for Partial Differential Equations (PDEs). In a recent work, we applied the DPG method with optimal test functions for the time integration of transient parabolic PDEs. We showed that the resulting DPG-based time-marching scheme is equivalent to exponential integrators for the trace variables. In this work, we extend the aforementioned method to time-dependent hyperbolic PDEs. For that, we reduce the second order system in time to first order and we calculate the optimal testing analytically. We also relate our method with exponential integrators of Gautschi-type. Finally, we validate our method for 1D/2D + time linear wave equation after semidiscretization in space with a standard Bubnov–Galerkin method. The presented DPG-based time integrator provides expressions for the solution in the element interiors in addition to those on the traces. This allows to design different error estimators to perform adaptivity.

Gas discharge mechanism of weakly ionized gas sensor using nanomaterials

Gas discharge plays a very important role in the performance of ionized gas sensor. Due to the structure of nanomaterial-based electrode, gas discharge becomes more complexity under atmospheric pressure. In this article, we show a theoretical framework for the calculation of gas discharge of weakly ionized gas sensor using nanomaterials. It consists of the mass conservation equation for each species with the drift-diffusion approximation, the electron energy conservation equation, and the Poisson’s equation. The effects of ionization, excitation, attachment, recombination, electron diffusion, and is formulated by a time-dependent finite-element partial differential equation solver. The simulation results agree well with experimental data under the same conditions and gas discharge characters of ionized gas sensor can be extracted from the model to understand the discharge mechanism. Our approach is not only suitable for explaining the experimental phenomena by clarifying the gas chemical reaction process but also to design the sensor structure.

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.

Methods to Recover Unknown Processes in Partial Differential Equations Using Data

We study the problem of identifying unknown processes embedded in time-dependent partial differential equation (PDE) using observational data, with an application to advection-diffusion type PDE. We first conduct theoretical analysis and derive conditions to ensure the solvability of the problem. We then present a set of numerical approaches, including Galerkin type algorithm and collocation type algorithm. Analysis of the algorithms are presented, along with their implementation detail. The Galerkin algorithm is more suitable for practical situations, particularly those with noisy data, as it avoids using derivative/gradient data. Various numerical examples are then presented to demonstrate the performance and properties of the numerical methods.

Approximate Controllability for Time-Dependent Impulsive Neutral Stochastic Partial Differential Equations with Fractional Brownian Motion and Memory

In this manuscript, we investigate the approximate controllability for time-dependent impulsive neutral stochastic partial differential equations with fractional Brownian motion and memory in Hilbert space. By using semigroup theory, stochastic analysis techniques and fixed point approach, we derive a new set of sufficient conditions for the approximate controllability of nonlinear stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate our results.

PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the method

To extend prevailing scaling limits when solving time-dependent partial differential equations, the parallel full approximation scheme in space and time (PFASST) has been shown to be a promising parallel-in-time integrator. Similar to space–time multigrid, PFASST is able to compute multiple time-steps simultaneously and is therefore in particular suitable for large-scale applications on high performance computing systems. In this work we couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator. While PFASST provides global, large-scale “parallelization across the step”, the inner parallel SDC method allows integrating each individual time-step “parallel across the method” using a diagonalized local Quasi-Newton solver. This new method, which we call “PFASST with Enhanced concuRrency” (PFASST-ER), therefore exposes even more temporal concurrency. For two challenging nonlinear reaction-diffusion problems, we show that PFASST-ER works more efficiently than the classical variants of PFASST and can use more processors than time-steps.

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.

Analysis and Performance Evaluation of Adjoint-guided Adaptive Mesh Refinement for Linear Hyperbolic PDEs Using Clawpack

Adaptive mesh refinement (AMR) is often used when solving time-dependent partial differential equations using numerical methods. It enables time-varying regions of much higher resolution, which can selectively refine areas to track discontinuities in the solution. The open source Clawpack software implements block-structured AMR to refine around propagating waves in the AMRClaw package. For problems where the solution must be computed over a large domain but is only of interest in a small area, this approach often refines waves that will not impact the target area. We seek a method that enables the identification and refinement of only the waves that will influence the target area. Here we show that solving the time-dependent adjoint equation and using a suitable inner product allows for a more precise refinement of the relevant waves. We present the adjoint methodology in general and give details on the implementation of this method in AMRClaw. Examples and a computational performance analysis for linear acoustics equations are presented. The adjoint method is compared to AMR methods already available in AMRClaw, and the advantages and disadvantages are discussed. The approach presented here is implemented in Clawpack, in Version 5.6.1, and code for all examples presented is archived on Github.

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.

Matrix Form of Deriving High Order Schemes for the First Derivative

For many problems in Physics and Computational Fluid Dynamics (CFD), providing an accurate approximation of derivatives is a challenging task. This paper presents a class of high order numerical schemes for approximating the first derivative. These approximations are derived based on solving a special system of equations with some unknown coefficients. The construction method provides numerous types of schemes with different orders of accuracy. The accuracy of each scheme is analyzed by using Fourier analysis, which illustrates the dispersion and dissipation of the scheme. The polynomial technique is used to verify the order of accuracy of the proposed schemes by obtaining the error terms. Dispersion and dissipation errors are calculated and compared to show the features of high order schemes. Furthermore, there is a plan to study the stability and accuracy properties of the present schemes and apply them to standard systems of time dependent partial differential equations in CFD.

Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.

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.