Numerical Solution Of Hyperbolic Equations Research Articles

Метод адаптивной искусственной вязкости для численного решения гиперболических уравнений и систем

Метод адаптивной искусственной вязкости для численного решения гиперболических уравнений и систем

A conservative implicit multirate method for hyperbolic problems

This work focuses on the improvement of a self-adjusting multirate strategy based on an implicit time discretization for the numerical solution of hyperbolic equations that allows to employ different time steps in different areas of the spatial domain. We propose a novel mass conservative multirate approach that can be generalized to various implicit time discretization methods. Mass conservation is achieved by flux partitioning, so that mass exchanges between a cell and its neighbors are exactly balanced. A number of numerical experiments on both nonlinear scalar problems and systems of hyperbolic equations have been carried out to test the efficiency and accuracy of the proposed approach.

A High Order Accurate Nodal Discontinuous Galerkin Method (DGM) for Numerical Solution of Hyperbolic Equation

A High Order Accurate Nodal Discontinuous Galerkin Method (DGM) for Numerical Solution of Hyperbolic Equation

Nonlinear data-bounded polynomial approximations and their applications in ENO methods

A class of high-order data-bounded polynomials on general meshes are derived and analyzed in the context of numerical solutions of hyperbolic equations. Such polynomials make it possible to circumvent the problem of Runge-type oscillations by adaptively varying the stencil and order used, but at the cost of only enforcing C 0 solution continuity at data points. It is shown that the use of these polynomials, based on extending the work of Berzins (SIAM Rev 1(4):624---627, 2007) to nonuniform meshes, provides a way to develop positivity preserving polynomial approximations of potentially high order for hyperbolic equations. The central idea is to use ENO (Essentially Non Oscillatory) type approximations but to enforce additional restrictions on how the polynomial order is increased. The question of how high a polynomial order should be used will be considered, with respect to typical numerical examples. The results show that this approach is successful but that it is necessary to provide sufficient resolution inside a front if high-order methods of this type are to be used, thus emphasizing the need to consider nonuniform meshes.

Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition

AbstractIn this work the combined finite difference and spectral methods have been proposed for the numerical solution of the one‐dimensional wave equation with an integral condition. The time variable is approximated using a finite difference scheme. But the spectral method is employed for discretizing the space variable. The main idea behind this approach is that we can get high‐order results. The new method is used for two test problems and the numerical results are obtained to support our theoretical expectations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

On an Implicit ENO Scheme

Essentially non-oscillatory (ENO) schemes, which have high order accuracy in regions where solutions are smooth and effectively capture shocks and discontinuities, have been developed for numerical solution of hyperbolic equations. These methods have traditionally been implemented using explicit Euler or explicit Runge-Kutta time marching schemes and consequently suffer from the Courant-Friedrichs-Lewy (CFL) time step restriction. This restriction is significant for steady-state problems and for transient problems with fronts whose speed is slow compared to the propagation speed in the rest of the domain. In this paper, a novel implicit time marching implementation of the ENO Roe scheme is developed. Simulation results which demonstrate the benefits of this implicit implementation are presented.

A fourier series method for the numerical solution of hyperbolic equations

In this paper a Fourier series technique which reduces the given hyperbolic partial differential equation to a system of ordinary differential equations with one point boundary conditions is presented. The numerical results obtained indicate a considerable saving in time in addition to accuracy since no discretisation errors are incurred in the space variable.

Numerical solution of hyperbolic equations for electron drift in strongly non-uniform electric fields

A comparison is made of solutions of the continuity equations for the motion of electrons and ions in a strong electric field using the methods of Euler, Runge-Kutta, Lax-Wendroff, characteristics and the flux-corrected transport (FCT) algorithm “Phoenical LPE Shasta” developed by Boris and Book( J. Comput. Phys. 20 (1976) , 397), with their flux limiter and with the new flux limiting algorithms developed by Zalesak ( J. Comput. Phys. 31 (1979) , 335). Results with and without ionization, and for uniform and non-uniform electric fields, are compared. Because ionization by electrons is strongly dependent on the electric field, considerable care needs to be taken to choose an optimum numerical scheme for non-uniform fields. For example, the calculation of the properties of corona discharges, or the cathode-fall of a glow discharge, requires fine spatial resolution. It is found that the Lax-Wendroff method and the method of characteristics give acceptable results; however, the Phoenical LPE Shasta algorithm with the flux limiting algorithm of Zalesak gives the best results, with the added advantage of suppressing spurious oscillations.

Extrapolation to the limit for numerical solutions of hyperbolic equations

The application of extrapolation to the limit requires the existence of an asymptotic expansion in powers of the step size. In this paper one-and multi-step methods for the solution of hyperbolic systems of first order are considered. Conditions are formulated that ensure the asymptotic expansion. Methods of characteristics for quasilinear systems with two independent variables are included in this presentation. If a rectangular grid is used, also non-quasilinear systems are admissible. The main part of this paper deals with initial value problems. But it is shown that in some exceptional cases asymptotic expansions hold for initial-boundary problems, too.

A boundary condition for significantly reducing boundary reflections with a Lagrangian mesh

Reflections from the terminating mesh boundaries are an important consideration in the selection of the calculational grid size for the numerical solution of hyperbolic equations. An adjusted damper system along the boundaries is the technique for significantly reducing the boundary reflections which are discussed here. Fixed, scattering, and damped boundary conditions are compared for a two-dimensional calculation of an elastic material subjected to a sinusoidally varying pressure. The damping condition resulted in reflections which were about 11 percent of the amplitude of the reflection from a fixed or free boundary, and about 33 percent of that from a scattering boundary. 4 figures. (RWR)

Higher Order Compact Implicit Schemes for the Wave Equation

Higher order finite-difference techniques have associated large star systems which engender complications near the boundary. In the numerical solution of hyperbolic equations, such boundary conditions require careful treatment since errors or instabilities generated there will in general pollute the entire calculation. To circumvent this difficulty, we use a class of implicit schemes suggested by H.-O. Kreiss, which achieves the highest order of accuracy possible on the smallest (most compact) mesh system. Here we develop a scheme which approximates the wave equation, \[ {U_{tt}} = a(x,y,t){U_{xx}} + b(x,y,t){U_{yy}},\] with fourth order accuracy in space and time. After an appropriate factorization, the resulting set of equations are tridiagonal and hence easily solved. The tridiagonal nature also indicates that the boundary conditions do not create special difficulties. Numerical experiments demonstrate the expected order of convergence and fulfill our expectations on the treatment of boundary conditions. An experimental computation also demonstrates that our results hold on L-shaped domains.

Higher order compact implicit schemes for the wave equation

Higher order finite-difference techniques have associated large star systems which engender complications near the boundary. In the numerical solution of hyperbolic equations, such boundary conditions require careful treatment since errors or instabilities generated there will in general pollute the entire calculation. To circumvent this difficulty, we use a class of implicit schemes suggested by H.-O. Kreiss, which achieves the highest order of accuracy possible on the smallest (most compact) mesh system. Here we develop a scheme which approximates the wave equation, \[ U t t = a ( x , y , t ) U x x + b ( x , y , t ) U y y , {U_{tt}} = a(x,y,t){U_{xx}} + b(x,y,t){U_{yy}}, \] with fourth order accuracy in space and time. After an appropriate factorization, the resulting set of equations are tridiagonal and hence easily solved. The tridiagonal nature also indicates that the boundary conditions do not create special difficulties. Numerical experiments demonstrate the expected order of convergence and fulfill our expectations on the treatment of boundary conditions. An experimental computation also demonstrates that our results hold on L-shaped domains.

The application of difference schemes of high accuracy to the numerical solution of hyperbolic equations

The application of difference schemes of high accuracy to the numerical solution of hyperbolic equations

An investigation, by the method of characteristics, of the lateral expansion of the gases behind a detonating slab of explosive

The phenomena occurring when an uncased explosive charge is detonated in a fluid medium are examined by hydrodynamical methods. Attention is focused chiefly on the pressure and velocity distributions in the gaseous products of the explosion, which expand laterally behind the detonation wave as it travels down the charge, the results being shown in graphical form. To simplify the problem, the charge, and the gas and fluid fields, were treated as two-dimensional. The hydrodynamical equations were solved numerically using the method of characteristics. This dates back to Monge, but it is only recently that it has been applied to the numerical solution of hyperbolic equations. The methods of numerical integration used in this paper are similar to those developed early in the war by the Research Section of the External Ballistics Department, Ordnance Board, for determining the velocity distributions around projectiles moving at supersonic speeds. The nature of the boundary conditions made it necessary to find explicit theoretical formulae for the gas field near the charge, and the analysis involved is given at length. For the problem in which the surrounding medium is air, the shape and position of the shock waves set up by the explosion are calculated. The shock waves are found to be straight to the nominal accuracy of the calculations (1 in 5000) for six charge widths from their intersections with the block of explosive.