A new efficient energy-preserving finite volume element scheme for the improved Boussinesq equation

A meshless finite point method for the improved Boussinesq equation using stabilized moving least squares approximation and Richardson extrapolation

“…Obviously, the FPM results are similar as those in References [10, 18, 22]. In fact, as stated in References [10, 18, 22], it can be observed from Table 5 that $$$ {A}_m $$$ is always smaller than the twice of $$$ \max \left({A}_1,{A}_2\right) $$$, that is, $$$ {A}_m<2\max \left({A}_1,{A}_2\right) $$$. More precisely, we can further find from Table 5 that $$$ {A}_m $$$ is always smaller than the sum of the two wave amplitudes $$$ {A}_1 $$$ and $$$ {A}_2 $$$, that is, $$$ {A}_m<{A}_1+{A}_2 $$$.…”

“…Besides, let $$$ K $$$ be the inelasticity coefficient, $$$$ K=\frac{A_m}{\max \left({A}_1,{A}_2\right)}. $$$$ The FPM results of $$$ {A}_m $$$ and $$$ K $$$ are listed in Table 5 and compared with those of the FDM [10], FVEM [18], and LDG [22] for various amplitudes $$$ {A}_1 $$$ and $$$ {A}_2 $$$. Obviously, the FPM results are similar as those in References [10, 18, 22].…”

“…Results of L ∞ error obtained using h = 0.1 and the FPM with 𝜏 = 0.01 and the FDMs [10,11], FEM [11], and FVEM [18] with 𝜏 = 0.001 for −80 < x < 140 and 0 < t ≤ 72.…”

“…The FPM results of A m and K are listed in Table 5 and compared with those of the FDM [10], FVEM [18], and LDG [22] for various amplitudes A 1 and A 2 . Obviously, the FPM results are similar as those in References [10,18,22].…”

“…Results of L ∞ , L 2 , and L 1 errors obtained by the FPM and FVEMs[17,18] for A = 0.50, −60 < x < 100, and 0 < t ≤ 1…”