When debris flows, landslides, or any gravitational mass flows hit closed or partially open water sources such as seas, oceans, fjords, hydraulic reservoirs, mountain lakes, bays, and landslide dams, it results in tsunami (impulse water waves) by transforming their impact energy to water body, potentially causing damages of infrastructures and human casualties both near field and the distant coastlines. The intensity of hazard depends on the scale, location and process of the landslide, and also on the reservoir volume and topography that surrounds it. Volume or size of the initial release mass that fails and slides along a slope is one of the dominant factors to determine the degree of splash, propagating speed and the amplitudes of the fluid waves, and potential dam breach or water spill over. Here, we numerically integrate the two-phase mass flow model [Pudasaini S. P., J. Geophysi. Res. 117(F03010), 2012] for quasi-three-dimensional, high-resolution simulation results with variation of size of the two-phase initial landslide or debris both longitudinally and laterally. In our numerical experimental results, we observe fundamentally different solid and fluid evolution and wave structures in the reservoir. There are also significant differences in the flow dynamics of solid under water for different volumes of the release mass by extending or contracting the base area along downslope and/or cross-slope directions. The simulation results show that tsunami amplitudes and run out extents are rapidly increased when the volume of the initial release mass in the form of a triangular wedge is increased by increasing the base area through the increment of the length and breadth of the release base. This study can be useful to develop and implement tsunami hazard mitigation strategies to enhance public safety and reduce potential loss due to landslide-generated wave hazards.

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Many physical problems in the real world are frequently modeled by ordinary differential equations (ODEs). Real-life problems are usually non-linear, numerical methods are therefore needed to approximate their solution. We consider different numerical methods viz., Explicit (Forward) and Implicit (Backward) Euler method, Classical second-order Runge-Kutta (RK2) method (Heun’s method or Improved Euler method), Third-order Runge-Kutta (RK3) method, Fourth-order Runge-Kutta (RK4) method, and Butcher fifth-order Runge-Kutta (BRK5) method which are popular classical iteration methods of approximating solutions of ODEs. Moreover, an intuitive explanation of those methods is also be presented, comparing among them and also with exact solutions with necessary visualizations. Finally, we analyze the error and accuracy of these methods with the help of suitable mathematical programming software.

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