Simulations and analysis of underwater acoustic wave propagation model based on Helmholtz equation

Figures & data

Figure 1. Model problem geometry region.

Figure 2. Example 1: Error behaviour in $L_2$ and $L_{\mathrm{\infty }}$ norm for N = 20 (left) and N = 30 (right).

Figure 3. Example 1: Numerical solution (left) vs exact solution (right) at N = 15 $\lambda =50$.

Figure 4. Example 1: Numerical solution (left) vs exact solution (right) at N = 30 $\lambda ={10}^2$.

Figure 5. Example 1: Numerical solution (left) vs exact solution (right) at N = 50 $\lambda ={10}^3$.

Figure 6. Example 1: Numerical solution (left) vs exact solution (right) at N = 100 $\lambda ={10}^4$.

Figure 7. Example 2: Numerical solution (left) vs exact solution (right) at N = 15 $\lambda =80$.

Figure 8. Example 2: Numerical solution (left) vs exact solution (right) at N = 30 $\lambda ={10}^2$.

Figure 9. Example 2: Numerical solution (left) vs exact solution (right) at N = 50 $\lambda ={10}^3$.

Figure 10. Example 2: Numerical solution (left) vs exact solution (right) at N = 100 $\lambda ={10}^4$.

Table 1. Example 2: Error behaviour between exact and numerical solution.

N	$L_2$ error	$L_{\mathrm{\infty }}$ error	N	$L_2$ error	$L_{\mathrm{\infty }}$ error
6	$2.283e^{-02}$	$6.137e^{-02}$	20	$1.489e^{-14}$	$3.977e^{-14}$
8	$8.384e^{-04}$	$2.543e^{-03}$	22	$2.138e^{-16}$	$6.031e^{-16}$
10	$9.218e^{-06}$	$2.806e^{-05}$	24	$2.898e^{-16}$	$8.368e^{-16}$
14	$5.723e^{-09}$	$1.512e^{-08}$	26	$2.924e^{-16}$	$8.507e^{-16}$
16	$1.312e^{-12}$	$2.676e^{-10}$	28	$2.836e^{-16}$	$7.533e^{-16}$
18	$1.300e^{-12}$	$1.516e^{-12}$	30	$4.364e^{-16}$	$1.606e^{-16}$

Figure 11. Example 3: Numerical solution (left) vs exact solution (right) at N = 10 $\lambda =15$.

Figure 12. Example 3: Numerical solution (left) vs exact solution (right) at N = 15 $\lambda =30$.

Figure 13. Example 3: Numerical solution (left) vs exact solution (right) at N = 20 $\lambda =40$.

Figure 14. Example 3: Numerical solution (left) vs exact solution (right) at N = 25 $\lambda =50$.

Table 2. Example 3: Error behaviour between exact and numerical solution.

N	$L_2$ error	$L_{\mathrm{\infty }}$ error	N	$L_2$ error	$L_{\mathrm{\infty }}$ error
6	$2.165e^{-01}$	$7.255e^{-01}$	20	$2.837e^{-08}$	$2.134e^{-07}$
8	$4.907e^{-02}$	$1.971e^{-02}$	22	$1.294e^{-09}$	$1.877e^{-08}$
10	$4.538e^{-03}$	$2.469e^{-03}$	24	$6.308e^{-11}$	$8.475e^{-09}$
14	$5.745e^{-05}$	$2.854e^{-04}$	26	$1.593e^{-11}$	$4.481e^{-10}$
16	$5.549e^{-06}$	$3.316e^{-05}$	28	$1.539e^{-12}$	$1.107e^{-11}$
18	$3.790e^{-07}$	$2.174e^{-06}$	30	$9.675e^{-12}$	$1.516e^{-12}$