Numerical Solutions of the 1D Convection–Diffusion–Reaction and the Burgers Equation Using Implicit Multi-stage and Finite Element Methods

Abstract

In the last decades, developments in computational mechanics motivated extensive research on numerical solutions that had an important impact on society [OdEtAl03]. In particular, we are interested in procedures that can be adapted to problems involving convective, diffusive, and reactive processes. These problems have a vast applicability (see [GoCoCa00], [TaShDe07], [KuEsDa04]), such as the simulation of pollution effects in rivers; modeling of the evolution of oil and natural gas reserves in the underground; modeling of heat transfer problems, dispersion of pollutants; modeling of cosmological scenarios, analysis in seismology; phenomenology of turbulence; the theory of shock waves; and in many other applications.Usually, the studies employ implicit multi-stage methods combined with the finite element method to increase the convergence region of the obtained results. In this discussion, we consider the implicit multi-stage method of second-order R 11 and fourth-order R 22 , for the discretization of the temporal domain and we use three formulations of the finite element method type for the discretization of the spatial domain, i.e., least squares (LSFEM), Galerkin (GFEM), and streamline-upwind Petrov-Galerkin (SUPG) to solve the 1D convection-diffusion-reaction and the Burgers equation.

A parameterized multi-step Newton method for solving systems of nonlinear equations

“…So, the m-step iterative method ATC has m+1 convergence order and it requires that the nonlinear function F(·) should have at least three Fréchet derivatives. Note that wide classes of important ODEs and PDEs, such as those arising in the Bratu problem, the Frank-Kamenetzkii problem [12], the Lene-Emden equation [13], the Burgers equation [14], the Klein-Gordon equation [15], the two-dimensional sinh-Poisson equation [16], and the three-dimensional nonlinear Poisson equation [17], heat equation, wave equation, Euler's beam equation etc., give rise to F(x) functions with high order Fréchet derivatives. Then, the multi-step iterative method ATC is applicable to wide classes of important problems.…”

“…Several numerical methods have been proposed recently for approximating the solution of (11)- (14) such as the homotopy method [19], the finite difference method [20,21], and the variational iteration method [22]. Also, Bao et al [23] suggested some high-accurate numerical methods for solving numerically (11)- (14).…”

“…Also, Bao et al [23] suggested some high-accurate numerical methods for solving numerically (11)- (14). Bao and Sun [24] applied a new technique based on time-splitting discretization for approximating the solution of a variant of (11)- (14).…”