The 2nd-order accurate total variation diminishing (TVD2) finite-difference method and p-version based space-time least-squares finite-element method (STLSFEM) are presented in this study. Both methods are applied to 1D unsteady Burgers' and shallow-water equations which contain sharp gradients. The computed results demonstrate that both methods are capable of resolving the sharp gradients without generating spurious oscillations. The methods provide the basis for the extension of TVD2 and p-version STLSFEM to multi-dimensional unsteady problems and appear to have great potential in computational fluid dynamics.
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