We consider a tsunami wave equation with singular coefficients and prove that it has a very weak solution. Moreover, we show the uniqueness results and consistency theorem of the very weak solution with the classical one in some appropriate sense. Numerical experiments are done for the families of regularised problems in one- and two-dimensional cases. In particular, the appearance of a substantial second wave is observed, travelling in the opposite direction from the point/line of singularity. Its structure and strength are analysed numerically. In addition, for the two-dimensional tsunami wave equation, we develop GPU computing algorithms to reduce the computational cost.

中文翻译：

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奇异地形的海啸传播

我们考虑一个具有奇异系数的海啸波动方程并证明它有一个非常弱的解。此外，我们在适当的意义上展示了非常弱解与经典解的唯一性结果和一致性定理。对一维和二维情况下的正则化问题族进行了数值实验。特别是，观察到大量第二波的出现，沿与奇点点/线相反的方向传播。对其结构和强度进行了数值分析。此外，对于二维海啸波动方程，我们开发了GPU计算算法来降低计算成本。