Abstract
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.
Highlights
A second order quasilinear Partial Differential Equations (PDEs) in two independent variables x, y is an equation of the form∂2U ( x, y) ∂2U ( x, y) ∂2U ( x, y) A ∂x2 +B +C ∂x∂y ∂y2 + D =0, (1)where A, B, C, D may be functions of x, y
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 problems for the one-dimension homogeneous wave equation
To test the method it is applied for the numerical solution of initial and boundary value problems (IBVPs) for the one-dimension homogeneous wave equation and it is compared with the following well-known finite difference methods: Central Time Central Space (CTCS), Crank-Nicolson and ω scheme
Summary
A second order quasilinear PDE in two independent variables x, y is an equation of the form. A modification of the numerical method of characteristics is proposed to solve special cases of initial and boundary value problems (IBVPs) for second order hyperbolic PDEs. To test the method it is applied for the numerical solution of IBVPs for the one-dimension homogeneous wave equation and it is compared with the following well-known finite difference methods: Central Time Central Space (CTCS), Crank-Nicolson and ω scheme. The common thing among the schemes CTCS, Crank-Nicolson and ω is that it is required to compute the solution on the first time step before they can be employed [20] In the literature, this is usually done by approximating the initial derivative condition by centered or forward second order divided difference without any specific preference [20]-[27]. A “slow” processor was chosen intentionally to distinguish time consuming methods
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