A new numerical method of the higher-order Godunov-type is proposed. The spatial profile within a grid is interpolated with the cubic spline. The time integration is represented by the time evolution of the value and the first spatial derivative of the quantity. This scheme is successfully applied to a linear wave propagation with less diffusion retaining monotonicity and to the nonlinear hydrodynamic equations. The extension to higher dimensions and to more general hyperbolic equations including coupled hyperbolic-parabolic equations is straightforward.
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Yabe et al. (1988) studied this question.
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