Hyperbolic PDEs

Leapfrog is used with apparent success to solve hyperbolic pdes [528], but was proved unconditionally unstable for parabolic pdes in 1950 [424]. Richardson had been lucky, in that the instabilities had not made themselves felt in his (pencil and paper) calculations, in the course of the few iterations he worked. 

S. S. Abarbanel and A. E. Chertock. Strict stability of high-order compact implicit finite-difference schemes the role of boundary conditions for hyperbolic PDEs, I. J. Comput. Phys., 160 42-66, 2000. 

Linear first order hyperbolic PDEs were solved analytically in chapter 8 (section 8.1.2). Linear and nonlinear first order hyperbolic PDEs can be solved numerically using numerical method of lines illustrated in chapter 5.2. Eirst order hyperbolic PDEs are usually specified with a boundary condition at x = 0 and an initial condition. In this chapter first order hyperbolic PDEs are solved in the domain... 

Numerical Method of Lines for Second Order Hyperbolic PDEs... 

First order hyperbolic PDEs were solved numerically in section 10.1.5. Second order hyperbolic PDEs are usually specified with boundary conditions at x = 0 and X = 1. In addition, initial conditions for both the dependent variable and its time derivative are specified. The methodology is very similar to numerical method of lines for parabolic PDEs described in chapter 5.2. The only difference is that instead of a system of first order ODEs, second order hyperbolic PDEs result in a system of second order ODEs. The resulting system of second order ODEs is solved numerically in time. The methodology is illustrated with the following examples. 

Equation (9.101) can be seen to be a special differential equation with one independent variable. The number of variables in the hyperbolic PDE has thus been reduced from two to one. Comparing Equation (9.101) with the generalized form of Bessel s ... 

The strong form of a hyperbolic PDE, that is, the governing equation, the boundary conditions (BC), the displacement-strain relation (DS) and the initial conditions (IC) are given as follows (for example, the equation of classical elasto-dynamics for a Hookean solid with mass density p and elasticity constant tensor D see, e.g., Gurtin 1972 Selvadurai 2000b) ... 

Therefore the weak form and the variational principle (i.e., the principle of the energy minimization) of the hyperbolic PDE are obtained as follows ... 

It is possible to combine the two equations in one hyperbolic second-order PDE. This has the property of finite wave speed, both boundary conditions at the entrance are easily calculable, and it accounts for some of the phenomena of unmixing. This is not the place to treat this model in detail and, indeed, it is still finding fruitful applications.5 Another method for a hyperbolic model is to be found in [173]. 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

In order to describe adequately the hydrodynamics of the experimental fixed bed reactor, it is necessary to take into account the axial dispersion in the mathematical model. The time dependent continuity equation including axial dispersion for a fixed bed reactor is given by a partial differential equation (pde) of the parabolic/hyperbolic class. These types of pde s are difficult to solve numerically, resulting in long cpu times. A way to overcome these difficulties is by describing the fixed bed reactor as a cascade of perfectly stirred tank reactors. The axial dispersion is then accounted for by the number of tanks in series. For a low degree of dispersion (Bo < 50) the number of stirred tanks, N, and the Bodenstein number. Bo, are related as N Bo/2 [8].The fixed bed reactor is now described by a system of ordinary differential equations (ode s). No radial gradients are taken into account and a onedimensional model is applied. Mass balances are developed for both the gas phase and the adsorbed phase. The reactor is considered to be isothermal. 

Due to the assumption of constant density, the volumetric flow rate does not change, and the model can be expressed by concentrations. The basic volume element is let to shrink and the hyperbolic partial differential equation (PDE) is obtained ... 

In the PDE toolbox Eqs. (6.147)-(6.149) are named elliptic, parabolic, and hyperbolic, respectively, regardless of the values of the coefficients and boundary conditions. 

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