A multivariate spectral quasi-linearization method for the solution of (2+1) dimensional Burgers’ equations

Abstract

In this paper, we introduce the multi-variate spectral quasi-linearization method which is an extension of the previously reported bivariate spectral quasi-linearization method. The method is a combination of quasi-linearization techniques and the spectral collocation method to solve three-dimensional partial differential equations. We test its applicability on the (2 + 1) dimensional Burgers’ equations. We apply the spectral collocation method to discretize both space variables as well as the time variable. This results in high accuracy in both space and time. Numerical results are compared with known exact solutions as well as results from other papers to confirm the accuracy and efficiency of the method. The results show that the method produces highly accurate solutions and is very efficient for (2 + 1) dimensional PDEs. The efficiency is due to the fact that only few grid points are required to archive high accuracy. The results are portrayed in tables and graphs.

A Novel Multivariate Spectral Local Quasilinearization Method (MV-SLQLM) for Modelling Flow, Moisture, Heat, and Solute Transport in Soil

“…From the inspection of Table 3, we observe that for each method, both the computational time and condition number increase with time, T but with small values recorded for the MV-SLQLM. The present results suggest that employing the LLM to decouple the quasilinearized equations saves more on computational time than solving it as a compact matrix system of quasilinearized equations (see [72,84]). Furthermore, in agreement with Hidayat et al [49], Table 3 shows that low computational effort of collocation is indeed guaranteed for schemes with Kronecker delta properties.…”

“…In Table 3, the numerical results based on running/CPU time and condition number of the coefficient matrices, obtained using the proposed method, are compared with those presented by the TV-SCM [84] and MV-SQLM [72] at two subinterval lengths ½0, T in t variable and using the same number of collocation points in the spatial and time domains.…”

“…Further, a Lagrange polynomial of order N, for example, at the Chebyshev-Gauss-Lobatto points L i ðvÞ, is defined as [72]…”

“…For simplicity and efficiency, we express equations ( 34), (35a), (35b), and (35c) in matrix form, using the Kronecker product, as [72]…”

“…Later, Mkhatshwa et al [71] proposed an overlapping multidomain spectral method to study conjugate problems of conduction and MHD free convection flow of nanofluids over flat plates. Dlamini and Magagula [72] proposed a multivariate SQLM (MV-SQLM) to solve 2D Burger's equations. Lekoko et al [73] and Goqo et al [74] applied the MV-SQLM to analyze pressure and heat distribution in a filter chamber and to study heat and mass transfer in a porous media saturated with nanofluid, respectively.…”