Quasilinear Hyperbolic Partial Differential Equations Research Articles

Riemann solutions of two-layered blood flow model in arteries

This study investigates the solutions of the Riemann problem for a two-layered blood flow model which is modeled by a system of quasi-linear hyperbolic partial differential equations (PDEs) obtained by vertically averaging the Euler equations over each layer. We explore the elementary waves, namely shock wave, rarefaction wave and contact discontinuity wave on the basis of method of characteristics. Further, we establish the existence and uniqueness of the corresponding local Riemann solution. Across the contact discontinuity wave, the areas of two nonlinear algebraic equations are determined by using the Newton–Raphson method of two variables in all possible wave combinations. A precise analytical method is used to display a detailed vision of the solution for this model inside a specified space domain and some certain time frame.

High resolution numerical algorithm of order (2, 4, 4) for 2D quasi-linear hyperbolic equations based on half-step discretization on an unequal grid

This study suggests a novel three-level implicit half-step scheme of order two and four in time and space direction respectively, utilizing an unequal grid for the approximation of 2D quasi-linear hyperbolic partial differential equations on an unequal grid. The stability analysis of the model Telegraphic equation for unequal grid has been discussed and it has been demonstrated that the presented method is unconditionally stable for the Telegraphic equation. The alternating direction implicit (ADI) techniques are discussed for linear difference equations on a dissimilar domain. The developed method investigated several well-known physical problems on an unequal mesh to demonstrate its accuracy and efficiency.

DYNAMIC MODEL OF THERMAL PROCESSES IN A LIQUID FLOWING THROUGH A CHANNEL

The article presents a dynamic thermal process model for an incompressible fluid moving through a channel of arbitrary cross-sectional shape. The study focuses on expressing the non-steady law of conservation of internal energy of the fluid by considering the dependence of parameters on both spatial coordinates and time. The one-dimensional spatial formulation of the problem is simplified by disregarding any changes in flow parameters in the direction perpendicular to the channel axis. The resulting equation represents a non-stationary, non-homogeneous, quasi-linear hyperbolic partial differential equation, which can be solved with the method of characteristics in the case of constant coefficients. The solution consists of two distinct solutions that are glued along the line x/t=v, with isolines shown in the figure provided. The research is relevant for the design of heat exchangers, rocket and aircraft engines, and other heat engineering devices.

Similarity solution for isothermal flow behind the magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas with the effect of the gravitational field

In this paper, we investigate a system of quasilinear hyperbolic partial differential equations, which describes the propagation of cylindrical shock waves in a rotating non-ideal gas with the effects of the gravitational field and the axial magnetic field. It is assumed that the flow is isothermal. The Lie group of transformations is used to generate the self-similar solutions of the considered problem in the medium of uniform density. The axial and azimuthal components of fluid velocity and magnetic field are supposed to be varying. We find the generators of the Lie group of transformations by employing the invariant surface criteria. We discovered four alternative solutions by selecting the arbitrary constants indicated in the generators' phrase. Only in three out of these four cases, the self-similar solutions exist. Two types of shock paths appear while solving the above cases. The power-law shock path appears in the first and third cases, while the exponential-law shock path appears in the second case. To find self-similar solutions, these cases have been solved numerically. We graphically show the distributions of flow variables behind the shock wave so that we can observe the effect on flow variables of the various values of the non-ideal parameter, Alfvén Mach number, adiabatic exponent, gravitational parameter, and ambient azimuthal velocity exponent. For the computational task, we used “MATLAB” software.

Numerical Method for 2D Quasi-linear Hyperbolic Equation on an Irrational Domain: Application to Telegraphic Equation

We develop a novel three-level compact method (implicit) of second order in time and space directions using unequal grid for the numerical solution of 2D quasi-linear hyperbolic partial differential equations on an irrational domain. The stability analysis of the model problem for unequal mesh is discussed and it is revealed that the developed scheme is unconditionally stable for the Telegraphic equation. For linear difference equations on an irrational domain, the alternating direction implicit method is discussed. The projected technique is scrutinized on several physical problems on an irrational domain to exhibitthe accuracy and effectiveness of the suggested method. Dhaka Univ. J. Sci. 69(2): 116-123, 2021 (July)

Nonlinear Wave Interactions in a Macroscopic Production Model

In this article, we study the exhaustive analysis of nonlinear wave interactions for a 2 × 2 homogeneous system of quasilinear hyperbolic partial differential equations (PDEs) governing the macroscopic production. We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations. Furthermore, we study the interaction between simple waves in detail through exact solution of general initial value problem. Finally, we discuss the all possible interaction of elementary waves using the solution of Riemann problem.

A Class of Second-Order Geometric Quasilinear Hyperbolic PDEs and Their Application in Imaging

Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent deve...

Nonclassical symmetry analysis, conservation laws of one-dimensional macroscopic production model and evolution of nonlinear waves

In this article we consider a system of first order quasilinear hyperbolic partial differential equations (PDEs) governing the one-dimensional macroscopic production model which describes high-volume product flows. We first show that the governing system admits nonlinear self-adjointness property. Further, we construct various conservation laws of the given system by using the direct multipliers method and also each of its classical symmetries yields a conservation law of the original system due to Ibragimov's formulation. Then we perform the nonclassical symmetry analysis and compute several hidden symmetries. Using these nonclassical symmetries, we obtain some new exact solutions and discuss their physical behavior graphically. Finally, the evolution of characteristic shock and weak discontinuity are studied by using one of the obtained solutions.

A new high accuracy cubic spline method based on half-step discretization for the system of 1D non-linear wave equations

PurposeThis paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D quasi-linear hyperbolic partial differential equations. The method is derived directly from the consistency condition of spline function which is fourth-order accurate. The method is directly applied to hyperbolic equations, irrespective of coordinate system, and fourth-order nonlinear hyperbolic equation, which is main advantage of the work.Design/methodology/approachIn this method, three grid points for the unknown function w(x,t) and two half-step points for the known variable x in spatial direction are used. The methodology followed in this paper is construction of a cubic spline polynomial and using its continuity properties to obtain fourth-order consistency condition. The proposed method, when applied to a linear equation is shown to be unconditionally stable. The technique is extended to solve system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the method.FindingsThe paper provides a fourth-order numerical scheme obtained directly from fourth-order consistency condition. In earlier methods, consistency conditions were only second-order accurate. This brings an edge over other past methods. In addition, the method is directly applicable to physical problems involving singular coefficients. Therefore, no modification in the method is required at singular points. This saves CPU time, as well as computational costs.Research limitations/implicationsThere are no limitations. Obtaining a fourth-order method directly from consistency condition is a new work. In addition, being an implicit method, this method is unconditionally stable for a linear test equation.Practical implicationsPhysical problems with singular and nonsingular coefficients are directly solved by this method.Originality/valueThe paper develops a new fourth-order implicit method which is original and has substantial value because many benchmark problems of physical significance are solved in this method.

Evolution of weak shock wave in two-dimensional steady supersonic flow in dusty gas

The present paper concerns with the propagation of weak shock waves in a dusty gas governed by the quasilinear hyperbolic partial differential equations. Using the method of wavefront analysis the transport equations governing the evolution of weak discontinuities are derived. Transport equations are used to determine the shock formation distance and conditions which ensure that there will not evolve any shock. Also the influence of dust particles, ratio of specific heats and upstream flow Mach number on the shock formation distance is analyzed.

A New Fast Algorithm Based on Half-Step Discretization for 3D Quasilinear Hyperbolic Partial Differential Equations

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.

A Computational Study with Finite Difference Methods for Second Order Quasilinear Hyperbolic Partial Differential Equations in Two Independent Variables

In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme Central Time Central Space (CTCS), Crank-Nicolson scheme, ω scheme and the method of characteristics for the numerical solution of initial and boundary value prob-lems for the one-dimension homogeneous wave equation. The initial deriva-tive condition is approximated by different second order difference quotients in order to examine which gives more accurate numerical results. The local truncation error, consistency and stability of the difference schemes CTCS, Crank-Nicolson and ω are also considered.

Online Model Identification Of Open-Channel System With High Order IDZ Model

In the control of open-channel, it is difficult to estimate wave propergation time because of the complexity of reflection, superposition and energy attenuation of shallow water waves. Although Saint-Venant equation have reasonable accuracy in describing the motion law of unsteady flow, mathematically it is the first order quasilinear hyperbolic partial differential equations which makes it difficult to be used as control model in the design optimized controller. The Channel Integrator Delay Zero (IDZ) model linearizes the Saint-Venant equation and ensures the accuracy of the response of water level and discharge in high frequency band. However, there is still a considerable difference between the theoretical value and the actual response. In order to avoid this difference caused by theoretical derivation, this paper uses the single channel model of Zhanghe Irrigation District to play online identification of the relevant parameters by using the existing periodic response process of the canal system. The reliability of model identification under step water intake is verified, and the effect of this method on periodic water intakes in long channel is verified. The results show that the identified model can catch most dynamic action of the canal system with water intakes, which ensures its validity to be used for controller design. Meanwhile, it is simple in application.

A Scalar Conservation Law for Plume Migration in Carbon Sequestration

A quasi-linear hyperbolic partial differential equation with a discontinuous flux models geologic carbon dioxide (CO$_2$) migration and storage [M. Hesse, F. Orr, and H. Tchelepi, Fluid Mech., 611 (2008), pp. 35--60]. Two flux functions characterize the model, giving rise to flux discontinuities. One convex flux describes the invasion of the plume into pore space, and the other captures the flow as the plume leaves CO$_2$ bubbles behind, which are then trapped in the pore space. We investigate the method of characteristics, the structure of shock and rarefaction waves, and the result of binary wave interactions. The dual flux property introduces unexpected differences between the structure of these solutions and those of a scalar conservation law with a convex flux. During our analysis, we introduce a new construction of cross-hatch characteristics in regions of the space-time plane where the solution is constant, and there are two characteristic speeds. This construction is used to generalize the notion of the Lax entropy condition for admissible shocks and is crucial to continuing the propagation of a shock wave if its speed becomes characteristic.

Investigation of the combined effect of air pockets and air bubbles on fluid transients

AbstractThis paper presents numerical and experimental investigations of the combined effect on pressure transients of air pockets and homogenous water–air bubble mixtures. An air pocket can accumulate at a high point of a pipeline along the control section located at the transition between pipes with sub- and supercritical slope, forcing open channel flow conditions underneath the pocket that ends in a hydraulic jump at the downward sloping pipe. The turbulence action at the jump generates small air bubbles that are entrained and transported along the pipe producing a two-component bubbly flow within the continuous liquid phase. A numerical model is developed, combining the explicit–implicit scheme proposed by McGuire and Morris and the method of characteristics for solving the quasi-linear hyperbolic partial differential equations for transient two-phase flow expressed in conservation form. To verify the proposed model, an experimental apparatus made of PVC was used to carry out hydraulic transient experiments. Tests were conducted in a tank–pipe–valve system and a valve with a pneumatic actuator at the downstream end generated transients. Numerical results at the test section pipe compares favorably with experimental data. The results show that pressure transients are significantly reduced with increasing air-pocket volumes and bubbly flow air content.

A New Spline-in-Tension Method of O( $$k^{2}+h^{4})$$ k 2 + h 4 ) Based on Off-step Grid Points for the Solution of 1D Quasi-linear Hyperbolic Partial Differential Equations in Vector Form

In this paper, we propose a new three level implicit method based on half-step spline in tension method of order two in time and four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form $$w_{tt}=K(x,t,w)w_{xx} + {\varphi }(x,t,w,w_{x},w_{t})$$ . We describe spline in tension approximations and its properties using two half-step grid points. The new method for one dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function w(x, t) and two half-step points for the known variable ‘x’ in x-direction. The proposed method when applied to Telegraphic equation is shown to be unconditionally stable. Further, the stability condition for 1-D linear hyperbolic equation with variable coefficients is established. Our method is directly applicable to hyperbolic equations irrespective of the coordinate system which is the main advantage of our work. The proposed method for scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems and numerical computations are provided to demonstrate the effectualness of the method.

Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations

The paper deals with the control and regulation by integral controllers forthe nonlinear systems governed by scalar quasi-linear hyperbolic partial differentialequations. Both the control input and the measured output are located on the boundary.The closed-loop stabilization of the linearized model with the designed integral controlleris proved first by using the method of spectral analysis and then by the Lyapunov directmethod. Based on the elaborated Lyapunov function we prove local exponential stabilityof the nonlinear closed-loop system with the same controller. The output regulationto the set-point with zero static error by the integral controller is shown uponthe nonlinear system. Numerical simulations by the Preissmann scheme are carriedout to validate the robustness performance of the designed controllerto face to unknown constant disturbances.

A New Fast Numerical Method Based on Off-Step Discretization for Two-Dimensional Quasilinear Hyperbolic Partial Differential Equations

We report a new 3-level implicit compact numerical method of order four in time and four in space based on off-step discretization for the solution of two-space-dimensional quasilinear hyperbolic equation [Formula: see text], [Formula: see text], [Formula: see text] defined in the region [Formula: see text], [Formula: see text], [Formula: see text]. We require only 19 grid points for the unknown variable [Formula: see text] and two extra off-step points each in [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to two-dimensional hyperbolic equations with singular coefficients, which is the main attraction of our work. We do not require any fictitious points for computation. The proposed method when applied to a two-dimensional damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve damped wave equation. Many benchmark problems are solved to confirm the fourth-order convergence of the proposed method.

A new spline in compression method of order four in space and two in time based on half-step grid points for the solution of the system of 1D quasi-linear hyperbolic partial differential equations

In this paper, we propose a new three-level implicit method based on a half-step spline in compression method of order two in time and order four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form u_{tt} =A(x,t,u)u_{xx} +f(x,t,u,u_{x},u_{t}). We describe spline in compression approximations and their properties using two half-step grid points. The new method for one-dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function u(x,t) and two half-step points for the known variable ‘x’ in x-direction. The proposed method, when applied to a linear test equation, is shown to be unconditionally stable. We have also established the stability condition to solve a linear fourth-order hyperbolic partial differential equation. Our method is directly applicable to solve hyperbolic equations irrespective of the coordinate system, which is the main advantage of our work. The proposed method for a scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.

A high resolution finite difference method for a model of structured susceptible‐infected populations coupled with the environment

We develop a general model describing a structured susceptible‐infected (SI) population coupled with the environment. This model applies to problems arising in ecology, epidemiology, and cell biology. The model consists of a system of quasilinear hyperbolic partial differential equations coupled with a system of nonlinear ordinary differential equations that represents the environment. We develop a second‐order high resolution finite difference scheme to numerically solve the model. Convergence of this scheme to a weak solution with bounded total variation is proved. Numerical simulations are provided to demonstrate the high‐resolution property of the scheme and an application to a multi‐host wildlife disease model is explored.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1420–1458, 2017