Abstract
In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.
Highlights
For the domain=Ω (0, L)× (0, L), let ∂Ω be the boundary of Ω and Ω = Ω ∂Ω
In [3] [4] upwind difference schemes were used to solve one-dimensional and two-dimensional hyperbolic equations with constant coefficient and stability analysis was conducted by Fourier method
Some high-order difference schemes were considered in [11] for first-order hyperbolic equations with variable coefficients, but the direction of flow rate was not taken into account
Summary
In [3] [4] upwind difference schemes were used to solve one-dimensional and two-dimensional hyperbolic equations with constant coefficient and stability analysis was conducted by Fourier method. Some high-order difference schemes were considered in [11] for first-order hyperbolic equations with variable coefficients, but the direction of flow rate was not taken into account. There is little work about the detailed discussions of upwind difference schemes for two- or three-dimensional hyperbolic equations with variable coefficients. In this paper, full-discrete explicit and implicit upwind difference schemes are formulated and analyzed for two-dimensional first-order hyperbolic equations with variable coefficients.
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