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首页> 外文期刊>International journal of modern physics, D. Gravitation, astrophysics, cosmology >Scalar hyperbolic equation with GR-type nonlinearity
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Scalar hyperbolic equation with GR-type nonlinearity

机译：具有GR型非线性的标量双曲方程

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摘要

We study a scalar hyperbolic partial differential equation, g(tt) = g(xx) - g(-1) (alphag(t)(2) + beta(g)(x)(2) + gammag(x)g(t)), with nonlinear terms similar to those of the equations of general relativity. The equation has a number of non-trivial analytical solutions whose existence rely on a delicate balance between linear and nonlinear terms. We formulate two classes of second-order accurate central-difference schemes, CFLN and MOL, for numerical integration of this equation. Solutions produced by the schemes converge to exact solutions at any fixed time t when numerical resolution is increased. However, in certain cases integration becomes asymptotically unstable when t is increased and resolution is kept fixed. This behavior is caused by subtle changes in the balance between linear and nonlinear terms when the equation is discretized. Changes in the balance occur without violating second-order accuracy of discretization. We thus demonstrate that a second-order accuracy, althoug necessary for convergence at finite t, does not guarantee a correct asymptotic behavior and long-term numerical stability. Accuracy and stability of integration are greatly improved by an exponential transformation of the unknown variable. [References: 6]
机译：我们研究了标量双曲偏微分方程，g（tt）= g（xx）-g（-1）（alphag（t）（2）+ beta（g）（x）（2）+ gammag（x）g（ t）），非线性项类似于广义相对论方程。该方程具有许多非平凡的解析解，它们的存在依赖于线性和非线性项之间的精细平衡。我们为方程的数值积分制定了两类二阶准确的中心差分方案，CFLN和MOL。当数值分辨率提高时，这些方案产生的解在任何固定时间t都收敛到精确解。但是，在某些情况下，当t增加并且分辨率保持固定时，积分会变得渐近不稳定。当方程离散化时，此行为是由线性项和非线性项之间的平衡的细微变化引起的。平衡的变化不会损害离散化的二阶精度。因此，我们证明了二阶精度（对于在有限t处收敛所需的所有精度）不能保证正确的渐近行为和长期数值稳定性。未知变量的指数变换极大地提高了积分的准确性和稳定性。 [参考：6]

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