Quasilinear Hyperbolic Equations Research Articles

Initial boundary value problem for the 3D quasilinear hyperbolic equations with nonlinear damping

ABSTRACTWe investigate global existence and asymptotic behavior of the 3D quasilinear hyperbolic equations with nonlinear damping on a bounded domain with slip boundary condition, which describes the propagation of heat waves for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of classical solutions are obtained when the initial data are near its equilibrium. Time asymptotically, the internal energy is conjectured to satisfy the porous medium equation and the heat flux obeys the classical Darcy’s-type law. Based on energy estimates, we show that the classical solution converges to steady state exponentially fast in time. Moreover, we also verify that the same is true for the corresponding initial boundary value problem of porous medium equation and thus justifies the validity of Darcy’s-type law in large time.

Iterative Approximate Factorization of Difference Operators of High-Order Accurate Bicompact Schemes for Multidimensional Nonhomogeneous Quasilinear Hyperbolic Systems

For solving equations of multidimensional bicompact schemes, an iterative method based on approximate factorization of their difference operators is proposed. The method is constructed in the general case of systems of two- and three-dimensional quasilinear nonhomogeneous hyperbolic equations. The unconditional convergence of the method is proved as applied to the two-dimensional scalar linear advection equation with a source term depending only on time and space variables. By computing test problems, it is shown that the new iterative method performs much faster than Newton’s method and preserves a high order of accuracy.

Kinetic and entropy solutions of quasilinear impulsive hyperbolic equations

In the present paper we deal with kinetic and entropy solutions of quasilinear impulsive hyperbolic equations. The genuine nonlinearity condition enables to prove the existence of one-sided traces of these solutions on fixed-time hyperplanes. The latter fact provides the impulsive condition. This type of equations can be used in the fluctuating hydrodynamics.

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.

The 3D quasilinear hyperbolic equations with nonlinear damping in a general unbounded domain

The 3D quasilinear hyperbolic equations with nonlinear damping in a general unbounded domain

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 sixth-order bicompact scheme with spectral-like resolution for hyperbolic equations

For the numerical solution of nonstationary quasilinear hyperbolic equations, a family of symmetric semidiscrete bicompact schemes based on collocation polynomials is constructed in the one- and multidimensional cases. A dispersion analysis of a semidiscrete bicompact scheme of six-order accuracy in space is performed. It is proved that the dispersion properties of the scheme are preserved on highly nonuniform spatial grids. It is shown that the phase error of the sixth-order bicompact scheme does not exceed 0.2% in the entire range of dimensionless wave numbers. A numerical example is presented that demonstrates the ability of the bicompact scheme to adequately simulate wave propagation on coarse grids at long times.

On one functional-differential quasilinear hyperbolic equation

For functional-differential quasilinear hyperbolic equation not investigated earlier, we consider problems with conditions on characteristics and obtain conditions of solvability.

Error analysis of implicit Runge–Kutta methods for quasilinear hyperbolic evolution equations

Error analysis of implicit Runge–Kutta methods for quasilinear hyperbolic evolution equations

Resonantly interacting non-linear waves in a van der Waals gas

The present paper uses the method of multiple time scales to derive the asymptotic solution of system of one-dimensional quasilinear hyperbolic equations for the generalized geometry in van der Waals gas. The transport equation for the amplitude of resonantly interacting high frequency waves propagating into non-ideal gas is derived. Further, we discuss the cases when the initial data for the wave amplitude is of 2π periodicity. The evolutionary behavior of non-resonant wave modes culminating into shock wave and its location are examined in van der Waals fluid.

Almost periodic solutions of evolution equations

We state existence theorems for almost periodic solutions of evolution problems, namely, quasi-autonomous problems and more generally, time dependent evolution equations. We apply these theorems firstly, to a boundary value quasilinear hyperbolic equation of first order, and secondly, to a boundary value quasi-parabolic equation.

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

Iterative approximate factorization for difference operators of high-order bicompact schemes for multidimensional nonhomogeneous hyperbolic systems

An iterative method for solving equations of multidimensional bicompact schemes based on an approximate factorization of their difference operators is proposed for the first time. Its algorithm is described as applied to a system of two-dimensional nonhomogeneous quasilinear hyperbolic equations. The convergence of the iterative method is proved in the case of the two-dimensional homogeneous linear advection equation. The performance of the method is demonstrated on two numerical examples. It is shown that the method preserves a high (greater than the second) order of accuracy in time and performs 3–4 times faster than Newton’s method. Moreover, the method can be efficiently parallelized.

Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of H3-norm. Furthermore, if, additionally, Lp-norm (1≤p<6/5) of the initial perturbation is finite, we also prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.