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On the discontinuous Galerkin computation of N ‐waves focusing
Author(s) -
Sescu Adrian,
Afjeh Abdollah A.
Publication year - 2010
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.2461
Subject(s) - total variation diminishing , mathematics , discontinuous galerkin method , mathematical analysis , discretization , classification of discontinuities , partial differential equation , conservation law , hyperbolic partial differential equation , nonlinear system , boundary value problem , boundary (topology) , upwind scheme , galerkin method , elliptic partial differential equation , finite difference method , computation , finite element method , physics , quantum mechanics , thermodynamics , algorithm
Sonic boom focusing phenomenon can be predicted using the solution to the nonlinear Tricomi equation which is a hybrid (hyperbolic‐elliptic) second‐order partial differential equation. In this paper, the hyperbolic conservation law form is derived, which is valid in the entire domain. In this manner, the presence of two regions where the equation behaves differently (hyperbolic in the upper and elliptic in the lower half‐plane) is avoided. On the upper boundary, a new mixed boundary condition for the acoustic pressure is employed. The discretization is carried out using a discontinuous Galerkin (DG) method combined with a Runge–Kutta total‐variation diminishing scheme. The results show the accuracy of DG methods to solve problems involving sharp gradients and discontinuities. Comparisons with analytical results for the linear case, and other numerical results using classical explicit and compact finite difference schemes and weighted essentially non‐oscillatory schemes are included. Copyright © 2010 John Wiley & Sons, Ltd.