Nonlinear Hyperbolic Partial Differential Equations Research Articles

Operator splitting method for simulation of dynamic flows in natural gas pipeline networks

We develop an operator splitting method to simulate flows of isothermal compressible natural gas over transmission pipelines. The method solves a system of nonlinear hyperbolic partial differential equations (PDEs) of hydrodynamic type for mass flow and pressure on a metric graph, where turbulent losses of momentum are modeled by phenomenological Darcy–Weisbach friction. Mass flow balance is maintained through the boundary conditions at the network nodes, where natural gas is injected or withdrawn from the system. Gas flow through the network is controlled by compressors boosting pressure at the inlet of the adjoint pipe. Our operator splitting numerical scheme is unconditionally stable and it is second order accurate in space and time. The scheme is explicit, and it is formulated to work with general networks with loops. We test the scheme over range of regimes and network configurations, also comparing its performance with performance of two other state of the art implicit schemes.

Modelling and simulation of pavement drainage

ABSTRACTThe development of a numerical model to describe pavement surface runoff by a system of non-linear hyperbolic partial differential equations enables the analysis and prediction of safety-relevant road design and construction. Aquaplaning risks could be minimised as well as damages to the road surface geometry prevented. The hydromechanical approach presented in this article by a two-dimensional Pavement Surface Runoff Model (PSRM) based on the depth-averaged Shallow Water Equations – in contrast to empirical models – represents a general simulation. PSRM includes features like the treatment of an irregular topography or the possibility to treat different surface textures. The model enables the calculation of water depths. Discontinuous solutions to the underlying equations are numerically gained by applying Riemann solvers with Godunov methods to compute approximate intercell fluxes over the discretised problem using Finite Volume Methods. The Harten–Lax–Van Leer (HLL) Riemann solver was chosen to calculate the intercell fluxes. Simulation examples show the wide range of possibilities to describe real pavement surface runoff problems, for example junctions and superelevation transitions. Flow resistance is modelled with the Darcy–Weisbach equation by considering the pavements mean texture depth (MTD) as a variable parameter. In order to take into account the pavement surface drainage capabilities due to infiltration processes in porous surface layers, a coupled three-dimensional numerical formulation enlarges the macroscopic PSR Model. The saturation of the porous surface geometry and the infiltration velocities by Darcy velocities under consideration of interface conditions on the interface between free flow on the road surface and drainage into the porous structure are implemented within an emerging hydromechanical numerical model.

Moving boundary problem for the detachment in multispecies biofilms

The work presents the qualitative analysis of the free boundary value problem related to the detachment process in multispecies biofilms. In the framework of continuum approach to one-dimensional mathematical modelling of multispecies biofilm growth, we consider the system of nonlinear hyperbolic partial differential equations governing the microbial species growth, the differential equation for the biomass velocity, the differential equation that governs the free boundary evolution and also accounts for detachment, and the elliptic system for substrate dynamics. The characteristics are used to convert the original moving boundary equation into a suitable differential equation useful to solve the mathematical problem. We also provide another form of the same equation that could be used in numerical applications. Several properties of the solutions to the free boundary problem are shown, such as positiveness of the functions that describe the microbial concentrations and estimates on the characteristic functions. Uniqueness and existence of solutions are proved by introducing a suitable system of Volterra integral equations and using the fixed point theorem.

On a free boundary problem for biosorption in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the biosorption process in multispecies biofilms. In the framework of continuum biofilm modeling, the mathematical problem consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear parabolic partial differential equations describing the substrate trends and a system of semilinear parabolic partial differential equations accounting for the diffusion, reaction and biosorption of different agents on the various biofilm constituents. Two systems of nonlinear hyperbolic partial differential equations have been considered as well for modeling the dynamics of the free and bounded sorption sites. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. The main result is the existence and uniqueness of the solutions to the stated free boundary value problem, which have been derived by converting the partial differential equations to Volterra integral equations and then using the fixed point theorem. Moreover, the work is completed with numerical simulations for a real case examining the growth of a heterotrophic–autotrophic biofilm devoted to wastewater treatment and acting as a sorbing material for heavy metal biosorption.

Neutral functional differential equations and systems of conservation laws

A natural way of introducing neutral functional differential equations is integration along the characteristics of the Riemann invariants of the hyperbolic partial differential equations. Up to now the problem has been discussed by A.D. Myshkis, K.L. Cooke and their followers in the linear or quasilinear case. The conservation laws are nonlinear hyperbolic partial differential equations whose study goes back to the fifties of the previous century, being due to the pioneering papers of O.A. Oleinik and P.D. Lax. These equations describe important phenomena in Physics and Engineering, being also subject to control issues. The present paper is an attempt to extend the method of integrating along the characteristics to the systems of conservation laws. The main purpose of this attempt is to emphasize the occurring nonlinear neutral functional differential equations and to construct the augmented validation (existence, uniqueness, continuous data dependence, Stability Postulate) at least at the level of the classical solutions.

Group Invariant Solutions of Nonlinear Time-Fractional Hyperbolic Partial Differential Equation

Group Invariant Solutions of Nonlinear Time-Fractional Hyperbolic Partial Differential Equation

Adaptive vibration control of a flexible marine riser via the backstepping technique and disturbance adaptation

This paper proposes an adaptive boundary control for vibration suppression of a flexible marine riser system. The dynamic model of the riser system is described in the form of a nonlinear nonhomogeneous hyperbolic partial differential equation and four ordinary differential equations. In a proper mathematical manner, the backstepping technique, Lyapunov’s direct method, and the adaptive technique are utilized to design an adaptive boundary control for the vibration suppression of the riser system, and also for the global stabilization of the riser within a small neighbourhood of its original position. In addition, a parameter adaptive law is designed to compensate for the system parametric uncertainties and a disturbance adaptation law is proposed to eliminate the effects of boundary disturbance. The uniformly bounded stability of the closed-loop riser system is achieved through rigorous Lyapunov analysis with no discretization or simplification of the partial differential equation dynamics model of the system. Simulation results are presented to illustrate the effectiveness of the proposed control.

Finite volume numerical solution to a blood flow problem in human artery

In this paper, we solve a one dimensional blood flow model in human artery. This model is of a non-linear hyperbolic partial differential equation system which can generate either continuous or discontinuous solution. We use the Lax–Friedrichs finite volume method to solve this model. Particularly, we investigate how a pulse propagates in human artery. For this simulation, we give a single sine wave with a small time period as an impluse input on the left boundary. The finite volume method is successful in simulating how the pulse propagates in the artery. It detects the positions of the pulse for the whole time period.

Clawpack: building an open source ecosystem for solving hyperbolic PDEs

Clawpack is a software package designed to solve nonlinear hyperbolic partial differential equations using high-resolution finite volume methods based on Riemann solvers and limiters. The package includes a number of variants aimed at different applications and user communities. Clawpack has been actively developed as an open source project for over 20 years. The latest major release, Clawpack 5, introduces a number of new features and changes to the code base and a new development model based on GitHub and Git submodules. This article provides a summary of the most significant changes, the rationale behind some of these changes, and a description of our current development model.

Qualitative analysis of the invasion free boundary problem in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the invasion model for multispecies biofilms. This model is based on the continuum approach for biofilm modeling and consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear elliptic partial differential equations describing the substrate trends and a system of semilinear elliptic partial differential equations accounting for the diffusion and reaction of motile species within the biofilm. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. By using the method of characteristics, the partial differential equations constituting the invasion model are converted to Volterra integral equations. Then, the fixed point theorem is used for the uniqueness and existence result. The work is completed with numerical simulations describing the invasion of nitrite oxidizing bacteria in a biofilm initially constituted by ammonium oxidizing bacteria.

Fuzzy-Model-Based Reliable Static Output Feedback $\mathscr{H}_{\infty }$ Control of Nonlinear Hyperbolic PDE Systems

This paper investigates the problem of output feedback robust $\mathscr{H}_{\infty }$ control for a class of nonlinear spatially distributed systems described by first-order hyperbolic partial differential equations (PDEs) with Markovian jumping actuator faults. The nonlinear hyperbolic PDE systems are first expressed by Takagi–Sugeno fuzzy models with parameter uncertainties, and then, the objective is to design a reliable distributed fuzzy static output feedback controller guaranteeing the stochastic exponential stability of the resulting closed-loop system with certain $\mathscr{H}_{\infty }$ disturbance attenuation performance. Based on a Markovian Lyapunov functional combined with some matrix inequality convexification techniques, two approaches are developed for reliable fuzzy static output feedback controller design of the underlying fuzzy PDE systems. It is shown that the controller gains can be obtained by solving a set of finite linear matrix inequalities based on the finite-difference method in space. Finally, two examples are presented to demonstrate the effectiveness of the proposed methods.

A path-conservative Osher-type scheme for axially symmetric compressible flows in flexible visco-elastic tubes

Flexible tubes are widely used in modern industrial hydraulic systems as connections between different components like valves, pumps and actuators. For the design and the analysis of the temporal behavior of a hydraulic system, one therefore needs an accurate mathematical model that describes the fluid flow in a compliant duct. Hence, in this paper we want to model the fluid–structure-interaction (FSI) problem given by the axially symmetric flow of a compressible barotropic fluid that flows through flexible tubes made of vulcanized rubber. The material of the tube can be described by using a visco-elastic rheology, which takes into account the strain relaxation of the material. The resulting mathematical model consists in a one-dimensional system of nonlinear hyperbolic partial differential equations (PDE) with non-conservative products and algebraic source terms. To solve this system numerically, we apply the DOT method, which is a generalized path-conservative Osher-type Riemann solver for conservative and non-conservative hyperbolic PDE recently proposed in [23] and [22].We provide numerical evidence that the proposed DOT Riemann solver is well-balanced for the governing PDE system under consideration. The method is compared to available quasi-exact solutions of the Riemann problem in the case of an elastic wall described by the Laplace law. It is also compared to available experimental data and exact solutions obtained in the frequency domain for a linear visco-elastic wall behavior. In all cases under investigation the proposed path-conservative finite volume scheme based on the DOT Riemann solver is able to produce very accurate results.

Qualitative analysis of the moving boundary problem for a biofilm reactor model

The work presents the qualitative analysis of the free boundary value problem related to a biofilm reactor model. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of nonlinear hyperbolic partial differential equations governing the microbial species growth, a system of semilinear elliptic partial differential equations describing the substrate trends, and a system of nonlinear differential equations for the mass balance in the reactor. The free boundary evolution is governed by a differential equation that also accounts for detachment and attachment. The main result is a uniqueness and existence theorem. By using the method of characteristics, the original differential system is converted to Volterra integral equations and then the fixed point theorem is used. The work is completed with numerical simulations describing the free boundary evolution and mainly focused on attachment/detachment process.

Weak and Young Measure Solutions for Hyperbolic Initial Boundary Value Problems of Elastodynamics in the Orlicz--Sobolev Space Setting

We establish the existence result for a global weak solution (respectively, Young measure solution) in the Orlicz--Sobolev space setting for the nonlinear hyperbolic initial boundary value problem to $u_{tt}=\text{div}(\sigma(Du))+\mu(\triangle u)_t$ (respectively, $u_{tt}=\text{div}(\sigma(Du))$), where the function $\sigma=\partial W/\partial F$ is continuous and the stored-energy function $W\colon \mathbb{M}^{d\times n}\to \mathbb{R}$ may be nonconvex. Our study is motivated by one-dimensional elastodynamics. The present paper gives first solvability results for nonlinear hyperbolic partial differential equations with nonpower-growth-type nonlinearity for $Du$ in the monotonicity case and in the case with lack of monotonicity. The results are new even for the one-dimensional case with $\sigma(\tau)=\ln^q(1+\vert{\tau}\vert)\vert{\tau}\vert^{p-2}\tau+a\tau$ for $q>0$ and $p\geq2$ and $a\in \mathbb{R}$; here $a>0$ corresponds to the strong ellipticity of $W$, $a=0$ --- the convexity of $W$, and $a<0$ ---...

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

In Vuik and Ryan (2014) we studied the use of troubled-cell indicators for discontinuity detection in nonlinear hyperbolic partial differential equations and introduced a new multiwavelet technique to detect troubled cells. We found that these methods perform well as long as a suitable, problem-dependent parameter is chosen. This parameter is used in a threshold which decides whether or not to detect an element as a troubled cell. Until now, these parameters could not be chosen automatically. The choice of the parameter has impact on the approximation: it determines the strictness of the troubled-cell indicator. An inappropriate choice of the parameter will result in detection (and limiting) of too few or too many elements. The optimal parameter is chosen such that the minimal number of troubled cells is detected and the resulting approximation is free of spurious oscillations. In this paper we will see that for each troubled-cell indicator the sudden increase or decrease of the indicator value with respect to the neighboring values is important for detection. Indication basically reduces to detecting the outliers of a vector (one dimension) or matrix (two dimensions). This is done using Tukey's boxplot approach to detect which coefficients in a vector are straying far beyond others (Tukey, 1977). We provide an algorithm that can be applied to various troubled-cell indication variables. Using this technique the problem-dependent parameter that the original indicator requires is no longer necessary as the parameter will be chosen automatically.

Mathematical Modeling of Heavy Metal Biosorption in Multispecies Biofilms

AbstractThe biofilm matrix is a complex of secreted polymers, absorbed nutrients and metabolites, cell lysis products, and even particulate material. Being polyanionic in nature, this matrix plays a crucial role in the biosorption of metal cations. In this work, the adsorption process of heavy metals in biofilms is modeled in one space dimension. The mathematical model is a free-boundary value problem for nonlinear hyperbolic and parabolic partial differential equations. Biomass and extracellular polymeric substances (EPS) growth is governed by hyperbolic equations, and substrate evolution by parabolic equations. All equations are mutually connected. The model is general and can work for any number of microbial species, EPS, and substrates. In numerical analysis, heterotrophic–autotrophic competition for space with oxygen as common substrate is considered. The model can describe biofilm growth dynamics including spatial distribution of microbial species, substrate concentrations, EPS formation, and, in pa...

Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients

Purpose – The purpose of this paper is to present the computational modeling of second-order two-dimensional nonlinear hyperbolic equations by using cosine expansion-based differential quadrature method (CDQM). Design/methodology/approach – The CDQM reduced the equations into a system of second-order differential equations. The obtained system is solved by RK4 method by converting into a system of first ordinary differential equations. Findings – The computed numerical results are compared with the results presented by other workers (Mohanty et al., 1996; Mohanty, 2004) and it is found that the present numerical technique gives better results than the others. Second, the proposed algorithm gives good accuracy by using very less grid point and less computation cost as comparison to other numerical methods such as finite difference methods, finite elements methods, etc. Originality/value – The author extends CDQM proposed in (Korkmaz and Dağ, 2009b) for two-dimensional nonlinear hyperbolic partial differential equations. This work is new for two-dimensional nonlinear hyperbolic partial differential equations.

Grid adaptation and non-iterative defect correction for improved accuracy of numerical solutions of PDEs

In this work we present a computational approach for improving the order of accuracy of a given finite difference method for solution of linear and nonlinear hyperbolic partial differential equations. The methodology consists of analysis of leading order terms in the discretization error of any given finite difference method, leading to a modified version of the original partial differential equation. Singular perturbations of this modified equation are regularized using an adaptive grid distribution and a non-iterative defect correction method is used to eliminate the leading order, regular perturbation terms in the modified equation. Implementation of this approach on a low order finite difference scheme not only results in an increase in its order of accuracy but also results in an improvement in its numerical stability due to the regularization of singular perturbations. The proposed approach is applied to four different canonical problems including the numerical solution of (1) Liouville equation, (2) inviscid Burgers equation, (3) nonlinear reaction–advection equation and (4) a system of hyperbolic PDEs. When compared to exact solutions, the numerical results demonstrate the ability of this method in both boosting the accuracy of finite difference schemes up to the desired order and also providing a fully stable numerical solution.

Computational Modeling of Discrete Mechanical Systems and Complex Networks: Where We are and Where We are Going

WHERE WE ARE IN MODELING MECHANICAL, TRAFFIC, AND SOCIO-ECONOMICAL DISCRETE SYSTEMS Discrete systems have been firstly introduced in physics (Fisher and Wiodm, 1969; Noor and Nemeth, 1980; Adhikari et al., 1996; Kornyak, 2009) to simulate materials at the microand nano-scales where a continuum description of matter breaks down. The constituents can be atoms or molecules and their interactions are usually modeled by force fields resulting from chemical potentials or weak van der Waals interactions, depending on the type of bonding. These models have been further exploited in mechanics with the aim of predicting macroscopic properties such as strength and toughness from the non-linear interactions taking place between the constituents at the different scales. Pioneering attempts in mechanics to model discrete mechanical systems are those using lattice models (van Mier et al., 1995; Schlangen and Garboczi, 1997) characterized by a network of nodes connected by links modeled by beams. Although proven to suffer from meshdependency, they have been broadly used for studying the non-linear fracture behavior of concrete at the meso-scale. Another line of research regards the generalizedBorn approach (Pellegrini and Field, 2002; Marenich et al., 2013), firstly used in chemistry to model a solute represented as a set of three-dimensional spheres into a continuum medium solvent, then applied in molecular mechanics (called MM/GBSA) to investigate contact and fracture of bodies at the micro-scale. The high-computational power and the development of powerful open source software allow nowadays the design of wide discrete scalable models composed of up to millions of particles or molecules whose equations of motions and mutual interactions are described by highly nonlinear interatomic potential laws. This is the field of molecular dynamics (MD), which led to the development of specific explicit time integration schemes (like the velocity-verlet integration scheme) to solve large systems of equations with a reduced computational cost. Car and Parrinello (1985) proposed a minimization of the total energy of the system by applying a dynamical simulated annealing based on MD. MD computations can also be coupled with continuum simulations by multi-scale methods. Among the many strategies available in the literature, (Shenoy et al., 1999; Knap and Ortiz, 2001) developed an approach based on the Tadmor’s quasi-continuum method (Tadmor et al., 1996) operating on a representative atomistic zone with a reduced number of degrees of freedom. Local minima of the whole system potential energy are determined via the total energy from a cluster of atoms, avoiding the complete calculation of the full atomistic force field. The MD enriched continuum method by Belytschko and Xiao (2003) and Xiao and Belytschko (2003) was also another pioneering approach to couple a potential energy Hamiltonian calculation conducted on a fine scale MD domain with a Lagrangian calculation on a coarse scale continuum domain with an overlapped bridging domain among the two representations. Recently, an implementation of interatomic potential laws within a displacement-based finite element (FE) formulation has also been proposed in Nasdala et al. (2010), with a rigorous implicit solution scheme, aiming at generating models where non-linear discrete and continuous systems can be suitably combined. Discrete systems made of nodes and links are also used in other disciplines than mechanics to model transport or socio-economical networks (Caldarelli and Vespignani, 2007; Whrittle, 2012). Based on a continuum version of traffic conservation along a link, Lighthill and Whitham (1955) and Whitham (1974) and independently Richards (1956) proposed the LWR model where the governing equation describing the dependency of the traffic flux on time and on location along a link is a nonlinear hyperbolic partial differential equation (PDE) analogous to that describing the propagation of the front of a wave inside a medium. For a homogenous link where the velocity of traffic is the same at any location and no shocks on the traffic flow are present, the integration of the LWR PDE leads to a non-linear relation between the traffic flow and the density of vehicles, which fully represents the traffic state. Also, in economics, it is of great interest to quantify the effect of breaking a link over the whole network response by simulating the dynamics of flow redistribution. Again a flow model can be used as suggested in Zhou et al. (2010) to decode a huge human crowd without distinction between

Accuracy and security analysis of transient flows in relatively long pipelines

Consideration is given, in this paper, to the accurate numerical solution of transient flows in relatively long pipelines caused by the water hammer phenomenon. The governing equations for such flows are a set of two coupled non-linear hyperbolic partial differential equations where the friction factor is a very important parameter. In these equations, the gravity force is introduced to take into account the pipeline inclination. The mathematical equations are solved in the time domain by the method of characteristics using linear integration. To reproduce correctly the transient flow, very small time increments were used in the constructed computer program. This program permits to get some systematic indications on the evolution and the damping of the pressure head waves due to a fast closing valve at the downstream end of a long pipeline. The study shows that results are different from those of short pipelines and small viscous fluids. To check the validity of the numerical model, computed results have been successfully compared with those found in the relevant literature. These results show that the gravity lift may have an important effect on the maximum pressures, which may become very important near the valve and cause failure of the pipeline especially in presence of defect. The safety factor, computed at equidistant sections of the line, determines the distance between the reservoir and the defect from which the failure may happen. For the considered application, results show that the pipeline is safe near the supply upstream tank.