Non‐oscillatory shock‐capturing finite element methods for the one‐dimensional compressible Euler equations

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Abstract
A class of shock‐capturing Petrov–Galerkin finite element methods that use high‐order non‐oscillatory interpolations is presented for the one‐dimensional compressible Euler equations. Modified eigenvalues which employ total variation diminishing (TVD), total variation bounded (TVB) and essentially non‐oscillatory (ENO) mechanisms are introduced into the weighting functions. A one‐pass Euler explicit transient algorithm with lumped mass matrix is used to integrate the equations. Numerical experiments with Burgers' equation, the Riemann problem and the two‐blast‐wave interaction problem are presented. Results indicate that accurate solutions in smooth regions and sharp and non‐oscillatory solutions at discontinuities are obtainable even for strong shocks.