## 1 Introduction  
Long wave run-up is difficult to observe in nature in real-time due to the catastrophic consequences it usually leads to. An illustration of a long wave is a tsunami, which has a wavelength far greater than the water depth and a small amplitude in the open ocean. When a tsunami reaches the nearshore region, its amplitude increases significantly, while its propagation speeds decreases. A large part of the theoretical results for run-up are based on shallow water theory. Analytical linear solutions for periodic long waves on a slope can be found, for example, in [PERSON] and Ferrers [1], [PERSON] [2], and [PERSON] and Keller [3]. Tsumamis are known for their dramatic run-up height. From the data collection of post-tsunami surveys, extreme run-up values have been observed that exceed 20m and the most severe damages were not always caused by the first wave [4; 5; 6]. Extreme run-up values cannot be explained by classical analytical solutions. A possible explanation for these extreme run-ups is resonance. [PERSON] [7] and [PERSON] [8] discussed the conditions for resonance in closed and open basins. The Helmholtz mode was also studied, which is the most important mode in natural basins and is normally observed in bays, inlets, and harbors with narrow entrance. [PERSON] [9] described the possibility of resonances in bays. [PERSON] and [PERSON] [10] introduced the concept of shelf resonance, which is also called quarter-wavelength resonance. [PERSON] et al. [11] found that reflection and refraction from the nearshore topographic features can lead to the amplification of run-up values of non-leading tsunami waves. Resonance effects in the process of run-up have been studied numerically by [PERSON] et al. [12; 13] and experimentally by [PERSON] et al. [14; 15]. Obviously boundary conditions play an important role in numerical simulations: they can easily make the difference between successful and unsuccessful computations, or between fast and slow ones. There is still no clear understanding of how to impose both incoming and outgoing waves at a boundary. One generally uses an open boundary condition seaward, which is a numerical artifact that lies in a limited computational domain to solve a mathematical problem that has no boundary at all. Such open boundary conditions are usually called \"radiation\", \"absorbing\", \"non-reflecting\", or \"far-field\" boundary conditions. They all allow waves to leave the truncated domain, thus avoiding spurious reflections that may pollute the solution in the interior of the computational domain of interest. They can be classified into two groups: non-reflecting boundary conditions and non-reflecting boundary layers. Non-reflecting boundary conditions absorb impinging waves on the artificial boundary. A classical example is the Sommerfeld boundary condition [16; 17]. On the other hand, non-reflecting boundary layers have the property of absorbing waves that are traveling inside the layer. Open boundaries are considered in a widespread literature. Examples are [PERSON] [18], [PERSON] and [PERSON] [19], [PERSON] and [PERSON] [20]. However, in a laboratory wave flume, not all reflections caused by a wave paddle can be eliminated. The resulting wave system is a multi-reflection system. In the framework of wave energy converters, [PERSON] et al. [21] discussed the re-reflection effects on the wave field. In nature, waves on some topographies are easily trapped, such as in narrow bays. When the water depth changes abruptly in shelves with piece-wise geometry, a multi-reflection system can also occur. Such reflective boundary conditions are seldom discussed. Run-up on a sloping beach with the use of an open boundary condition at the seaward boundary matches the linear analytical solution that shows no resonance at all. However, no matter if it is theory or experiment, there exist some resonant frequencies [2; 13; 22]. Wave trapping is one of the phenomena that lead to resonance [23; 24].