Abstract We study a symmetric three-level (in time) method with a weight and a symmetric vector two-level method for solving the initial-boundary value problem for a second-order hyperbolic equation with a small parameter $$\tau > 0$$ multiplying the highest time derivative, where the hyperbolic equation is a perturbation of the corresponding parabolic equation. It is proved that the solutions are uniformly stable in $$\tau $$ and time in two norms with respect to the initial data and the right-hand side of the equation. Additionally, the case where $$\tau $$ also multiplies the elliptic part of the equation is covered. The spacial discretization can be performed using the finite-difference or finite element method.

中文翻译：

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求解小参数二阶双曲方程数值方法的稳定性

摘要 我们研究了一种带权重的对称三水平（时间）方法和一种对称向量两水平方法，用于求解小参数 $$\tau > 0$ 的二阶双曲方程的初边值问题。 $ 乘以最高时间导数，其中双曲方程是对应抛物方程的扰动。证明了关于初始数据和方程右侧的解在 $$\tau $$ 和时间上在两个范数中是一致稳定的。此外，还涵盖了 $$\tau $$ 还乘以方程的椭圆部分的情况。可以使用有限差分或有限元方法进行空间离散化。