Numerical solutions for convection–diffusion equation using El-Gendi method

A fourth-order numerical scheme for solving the modified Burgers equation

“…Experiments proved that the most accurate results are obtained for h ≥ 0.001 and = 10 −5 . From the results given in Table 3 it is deduced that the method introduced gives more accurate results for all time levels used than the corresponding results in [37,38]-finally, better than [42] and of equivalent accuracy to those in [43,44]. In Fig.…”

“…The resulting nonlinear scheme was solved using an already known [47][48][49][50] MPC method in which the corrector, which is an explicit scheme, is applied once. The computational efficiency of the proposed method given in detail in Section 3 was tested by comparing the numerical results with the corresponding ones in [37,38,[42][43][44][45]. It was found that the method for the MBE2 equation gives more accurate results than [37,38] and ones of equivalent accuracy to those of [42][43][44], while for the MBE3 one [45] more accurate results are given at long time intervals.…”

“…Comparisons for the proposed method (h = 0.001 and = 10 −5 ). [43] 0.25975 0.07220 0.09872 0.03871 [44] 0.273 0.157 0.139 0.0936 Table 4 MBE2 equation. Results with h = 0.001, = 10 −5 and ρ = ρ 2,10 .…”

“…The MBE equation, which has the strong nonlinear aspects of the governing equation ( is used in many practical transport problems such as those of nonlinear waves in a medium with low-frequency pumping or absorption, ion reflection at quasi-perpendicular shocks, turbulence transport, wave processes in thermoelastic media, transport and dispersion of pollutants in rivers and sediment transport, etc. Theoretical solutions of the MBE equation are found in, among other references, Harris [33], Sachdev and Rao [34], Sachdev et al [35], Inan and Ugurlu [36], while numerical solutions are found in Ramadan and El-Danaf [37] using the collocation method with quintic splines, Ramadan et al [38] with finite elements with a method based on the collocation of septic B-splines, Dağ et al [39] with finite elements with a method based on the collocation of cubic B-splines, Dağ et al [40] using both quadratic and cubic B-spline Galerkin finite-element methods, Saka and Dağ [41] with the use of a quartic B-spline collocation method, Saka and Dağ [42] with the use of a quintic B-spline collocation method, Irk [43] with the sextic B-spline collocation method, Temsah [44] using the El-Gendi method, Duan et al [45] using a lattice Boltzmann model, etc. The initial condition associated with Eq.…”

“…For reasons of comparison with the corresponding work in[37,38,[42][43][44] the values -U n 0 = 0 and U N N+1 = u(1, t) with u(1, t) given by Eq. (3.2), and -ν = 0.01, 0.001 and 0.005 were used.…”