J Sci Comput (2017) 72:1232–1268
Construction of a p-Adaptive Continuous Residual
· Q. Viville
· H. Beaugendre
Received: 6 November 2015 / Revised: 12 February 2017 / Accepted: 21 February 2017 /
Published online: 13 March 2017
© Springer Science+Business Media New York 2017
Abstract A p-adaptive continuous residual distribution scheme is proposed in this paper.
Under certain conditions, primarily the expression of the total residual on a given element K
into residuals on the sub-elements of K and the use of a suitable combination of quadrature
formulas, it is possible to change locally the degree of the polynomial approximation of the
solution. The discrete solution can then be considered non continuous across the interface of
elements of different orders, while the numerical scheme still veriﬁes the hypothesis of the
discrete Lax–Wendroff theorem which ensures its convergence to a correct weak solution.
We detail the theoretical material and the construction of our p-adaptive method in the frame
of a continuous residual distribution scheme. Different test cases for non-linear equations at
different ﬂow velocities demonstrate numerically the validity of the theoretical results.
Keywords p-Adaptation · Continuous and discontinuous ﬁnite elements · Residual
distribution schemes · Non linear equations
Because of their potential in delivering higher accuracy with lower cost than low order meth-
ods, high-order methods for computational ﬂuid dynamics (CFD) have obtained considerable
attention in the past two decades . By high order, we mean third order or higher. Most
industrial codes used today for CFD simulations are based upon second-order ﬁnite volume
methods (FVM), ﬁnite difference methods (FDM), or ﬁnite element methods (FEM) .
As second-order methods are used in most CFD codes, some very complex ﬂows simu-
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Institut de Mathématiques de Bordeaux, 33 405 Talence, France
Inria Bordeaux Sud Ouest, 33 405 Talence, France
Institut Polytechnique de Bordeaux, 33 405 Talence, France