A High-Order Accurate Unstructured GMRES Solver for the Compressible Euler Equations

On Obtaining High-Order Finite-Volume Solutions to the Euler Equations on Unstructured Meshes

“…We speculate that there are two reasons for this: first, developing a genuinely high-order accurate flow solver requires attention to a large number of details, many of which are not documented in the literature; and second, high-order schemes are perceived as requiring prohibitive CPU time and/or memory. This paper will address the first of these issues by relating our experience 4,[13][14][15][16][17][18] in developing and demonstrating a high-order accurate solver for the Euler equations. The issue of efficiency, on which we have also made significant progress, is the topic of a companion paper.…”

A High-Order Accurate Unstructured Newton-Krylov Solver for Inviscid Compressible Flows

“…Newton-Krylov solvers [8][9][10][11][12][13][14] are used extensively in CFD simulations because of their property of semi-quadratic convergence when starting from a good initial solution. Since the GMRES algorithm, among other Krylov techniques, only needs matrix vector products and these products can be computed by matrix free approach, matrixfree GMRES 10 is a very practical technique for dealing with the complicated Jacobian matrices arising from higherorder discretization.…”

“…This approach saves memory usage considerably and removes the problem of explicitly forming the higher-order Jacobian matrix. Results of an unstructured mesh solver for Poisson's equation 13 clearly showed the possibility of reducing computational cost required for a given level of solution accuracy using higher-order methods and a non-preconditioned matrix free GMRES as a convergence acceleration technique. In case of problems with non-linear flux function, including CFD problems, the Jacobian is typically ill-conditioned and effective preconditioning is necessary for satisfactory GMRES convergence.…”

Identificação de parâmetros em problemas de advecção-difusão combinando a técnica do operador adjunto e métodos de volumes finitos de alta ordem

“…Assim sendo, uma proposta futura consiste em procurar um método de resolução de EDPs baseado em MVF com condições menos restritivas de estabilidade e que tenha uma boa taxa de convergência como por exemplo, o GMRES matrix-free [46,43,48]. O GMRES matrix-free, nos artigos citados, surge no contexto onde a EDPé resolvida implicitamente utilizando o MVF em malha não-estruturada de triângulos.…”