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[HDS10-04] A New Calculation Method of Wet/Dry Boundary for Tsunami Inundation -Part 2 Calculation of Advection Term at Wet/Dry Boundary
Keywords:tsunami, numeric calculation, inundation, boundary conditions
1 Background
In the tsunami inundation calculation, the inundation phenomenon is generally treated as a boundary condition between a mesh where water exists (hereinafter referred to as wet) and a mesh where no water exists (hereinafter referred to as dry). The method of Kotani et al. (1998) (Hereinafter referred to as Kotani's method), which is an improved version of the method of Iwasaki and Mano (1979), has been widely used in tsunami numerical calculations because of its superior calculation accuracy. The details of Kotani's method are as follows. (1) Runup calculation is performed only when the wave height on the wet side is higher than the ground height on the dry side. (2) The total water depth D when calculating the wet/dry boundary is the difference between the wave height on the wet side and the ground height on the dry side. (3) When the D becomes zero or smaller than a certain lower limit in the calculation of the advection term, only the term with those total water depth as the denominator is omitted from the calculation. On the other hand, it has been pointed out that the calculation of the run-up tip may become unstable when Kotani's method is used (Nuclear Civil Engineering Committee, 2007). Therefore, in Minami (2021, SSJ Fall Meeting), I proposed a method to calculate the D for the wet/dry boundary in (2) above by simply additive averaging the D on the wet side and the dry side. It is shown that the proposed method does not produce numerical oscillations and improves the computational stability in the situation where the Kotani's method produces computational instability. In the method of Minami (2021), the condition of (1) was the same as that of Kotani's method. Therefore, the remaining condition (3) is discussed in this paper.
2 Calculation method
In the condition (3) of Kotani's method, the total water depth is the key to calculate the term, but since the staggered grid is used in the calculation, the calculation of the advection term at the wet/dry boundary can be divided into three cases as follows. First, when the upwind side is wet (the downwind side can be either wet or dry), the advection terms can be calculated as usual. Next, when both the upwind and downwind sides are dry, the flux is always zero. Finally, when the upwind side is dry, as shown in Fig.1. In Fig. 1, the term of upwind side at the point indicated by the blue arrow is calculated as zero because the total water depth in the red circle is zero in the condition (3) of Kotani's method. However, the total water depth is already zero at point (i+1), and the advection term is a spatial derivative, so the calculation is physically more reliable if the value of Δx used in the difference equation is half the value. Next, If Kotani's method is used to calculate the flux of the blue arrow in Fig. 2, the advection term will be calculated using the flux of the yellow arrow as the upwind value. However, the total water depth is zero at point (i+1) and the flux is also zero. This means that the blue and yellow arrows are physically discontinuity. So, again, if the upwind term is set to zero and Δx is halved, the calculation will be physically consistent. So, I propose a method in which condition (3) of Kotani's method is changed to "When calculating the advection term in the wet/dry boundary calculation, if the total water depth at the point of wave height where the upwind side is zero, the term on the upwind side is set to zero and Δx is halved.
3 Results and discussion
The results of the tsunami numerical calculation using the above method were compared with those of Kotani's method, and the results were almost the same in many cases. The reason for this was that the difference in the conditions only appeared in a few calculation steps immediately after a certain mesh became dry during an undertow. However, this condition will be true when the flooded mesh becomes dry due to undertow, in which case the waves are somewhat more likely to recede.
In the tsunami inundation calculation, the inundation phenomenon is generally treated as a boundary condition between a mesh where water exists (hereinafter referred to as wet) and a mesh where no water exists (hereinafter referred to as dry). The method of Kotani et al. (1998) (Hereinafter referred to as Kotani's method), which is an improved version of the method of Iwasaki and Mano (1979), has been widely used in tsunami numerical calculations because of its superior calculation accuracy. The details of Kotani's method are as follows. (1) Runup calculation is performed only when the wave height on the wet side is higher than the ground height on the dry side. (2) The total water depth D when calculating the wet/dry boundary is the difference between the wave height on the wet side and the ground height on the dry side. (3) When the D becomes zero or smaller than a certain lower limit in the calculation of the advection term, only the term with those total water depth as the denominator is omitted from the calculation. On the other hand, it has been pointed out that the calculation of the run-up tip may become unstable when Kotani's method is used (Nuclear Civil Engineering Committee, 2007). Therefore, in Minami (2021, SSJ Fall Meeting), I proposed a method to calculate the D for the wet/dry boundary in (2) above by simply additive averaging the D on the wet side and the dry side. It is shown that the proposed method does not produce numerical oscillations and improves the computational stability in the situation where the Kotani's method produces computational instability. In the method of Minami (2021), the condition of (1) was the same as that of Kotani's method. Therefore, the remaining condition (3) is discussed in this paper.
2 Calculation method
In the condition (3) of Kotani's method, the total water depth is the key to calculate the term, but since the staggered grid is used in the calculation, the calculation of the advection term at the wet/dry boundary can be divided into three cases as follows. First, when the upwind side is wet (the downwind side can be either wet or dry), the advection terms can be calculated as usual. Next, when both the upwind and downwind sides are dry, the flux is always zero. Finally, when the upwind side is dry, as shown in Fig.1. In Fig. 1, the term of upwind side at the point indicated by the blue arrow is calculated as zero because the total water depth in the red circle is zero in the condition (3) of Kotani's method. However, the total water depth is already zero at point (i+1), and the advection term is a spatial derivative, so the calculation is physically more reliable if the value of Δx used in the difference equation is half the value. Next, If Kotani's method is used to calculate the flux of the blue arrow in Fig. 2, the advection term will be calculated using the flux of the yellow arrow as the upwind value. However, the total water depth is zero at point (i+1) and the flux is also zero. This means that the blue and yellow arrows are physically discontinuity. So, again, if the upwind term is set to zero and Δx is halved, the calculation will be physically consistent. So, I propose a method in which condition (3) of Kotani's method is changed to "When calculating the advection term in the wet/dry boundary calculation, if the total water depth at the point of wave height where the upwind side is zero, the term on the upwind side is set to zero and Δx is halved.
3 Results and discussion
The results of the tsunami numerical calculation using the above method were compared with those of Kotani's method, and the results were almost the same in many cases. The reason for this was that the difference in the conditions only appeared in a few calculation steps immediately after a certain mesh became dry during an undertow. However, this condition will be true when the flooded mesh becomes dry due to undertow, in which case the waves are somewhat more likely to recede.