AbstractA method based on backward finite differencing in time and a least‐squares finite element scheme for first‐order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes theL2‐norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high‐order elements, computed solutions based on theL2‐method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weightedH1‐norm of the residual is proposed and leads to a successful scheme with high‐degree elements. Finally, a conservative least‐squares finite element method is also developed. Numerical results for two‐dimensional problems are given to demonstrate the shock resolution of the methods and compare dif
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