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Comparison of the ENATE approach and discontinuous Galerkin spectral element method in 1D nonlinear transport equations

Llorente V., Rubio G., Pascau A., Ferrer E., Arıcı M.

Computer Assisted Methods in Engineering and Science

2016

Vol. 23, no. 2/3

133--146

In this paper a comparison of the performance of two ways of discretizing the nonlinear convection-diffusion equation in a one-dimensional (1D) domain is performed. The two approaches can be considered within the class of high-order methods. The first one is the discontinuous Galerkin method, which has been profusely used to solve general transport equations, either coupled as the Navier-Stokes equations, or on their own. On the other hand, the ENATE procedure (Enhanced Numerical Approximation of a Transport Equation), uses the exact solution to obtain an exact algebraic equation with integral coefficients that link nodal values with a three-point stencil. This paper is the first of a thorough assessment of ENATE by comparing it with well established high-order methods. Several test cases of the steady Burgers’ equation with and without source have been chosen for comparison.

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

Costa R., Machado G. J., Clain S.

International Journal of Applied Mathematics and Computer Science

2015

Vol. 25, no. 3

529--537

A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for one-dimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is obtained by solving a matrix-free problem. An application in elasticity coupling the two operators is presented. We consider a beam subject to a combination of tensile and bending loads, where the main goal is the stress critical point determination for an intramedullary nail.