ACCURATE UPWIND METHODS FOR THE EULER EQUATIONS

A new class of piecewise linear methods for the numerical solution of the one dimensional Euler equations of gas dynamics is presented. Thes e methods are uniformly second-order accurate and can be considered as extensions of Godunov's scheme. With an appropriate definition of mon otonicity preservation for the case of linear convection, it can be sh own that they preserve monotonicity. Similar to Van Leer's scheme, the y consist of two key steps: a reconstruction step followed by an upwin d step. For the reconstruction step, a monotonicity constraint that pr eserves uniform second-order accuracy is introduced. Computational eff iciency is enhanced by devising a criterion that detects the ''smooth' ' part of the data where the constraint is redundant, The concept and coding of the constraint are simplified by the use of the median funct ion. A slope-steepening technique, which has no effect at smooth regio ns and can resolve a contact discontinuity in four cells, is described . As for the upwind step, existing and new methods are applied in a ma nner slightly different from those in the literature. These methods ar e derived by approximating the Euler equations via linearization and d iagonalization. At a ''smooth'' interface, for economy, the linearizat ion employs the arithmetic average of the left and right states,and sh e Aux is obtained by one of the four simple models: the primitive-vari able splitting, a simplified flux splitting. the one-intermediate-stat e model, or the central difference with artificial viscosity. Near a d iscontinuity, Roe's flux-difference splitting is used. The current pre sentation of Roe's method, via the conceptually simple flux-vector spl itting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional stat ement and an efficient approximation to Osher's approximate Riemann so lver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions and yield high resolution at discontinuities.