A class of stable least‐square finite element methods for non‐linear hyperbolic problems is developed and some exploratory studies made. The methods are based on modifying the L 2 ‐norm of the. residual and a related approximation to the H 1 ‐norm of the residual. The effect of the additional terms in these residual functionals is to introduce a dissipative effect proportional to the solution gradient. This acts to stabilize the solution for non‐linear hyperbolic problems which generate shocks. Numerical results for a one‐dimensional nozzle and shock tube problem demonstrate the accuracy and stability of the method. Results are for an implicit scheme and calculations for linear, quadratic and cubic elements are given.
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Jiang et al. (1988) studied this question.
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