Finite element non-oscillatory schemes for compressible flows.

*(English)*Zbl 0702.76074
Pubbl., Ist. Anal. Numer. Cons. Naz. Ric., Pavia 730, 183-198 (1989).

The principal motivation of our previous and present works is to develop schemes to be used in the numerical simulation of compressible inviscid flows of perfect gas in two- or three-dimensional space. These schemes are also intended to be used for more general problems like viscous flows, hypersonic reactive gaseous mixtures, etc.

Among the difficulties one can meet and beside the classical stability problems are firstly, preserving positiveness for some physically significant variables, like pressure, density, temperature, species concentration and secondly, avoiding spurious oscillations.

The basic choices we made are: (1) the use of triangular (tetrahedral) meshes for geometric flexibility, (2) finite-volume formulation for conservativeness and (3) upwinding in order to approximate well the convective terms of the equations. All these ingredients will result in a finite-volume/finite-element conservative upwind scheme.

The basic upwind scheme is known to be monotone in the case of a one- dimensional scalar equation, but then, it is only first-order accurate. A second class of schemes were then derived from the ideas of preserving monotony and of stability conditions for the nonlinear scalar equation. These schemes are known as total variation diminishing schemes (T.V.D.). The most interesting T.V.D. schemes are the nonlinear ones, as linearity would imply monotony and thus first-order accuracy.

This led to constructing new schemes or more often developing techniques for transforming existing schemes (that is, first-order upwind schemes, centered schemes, F.E.M Ritz-Galerkin...) into higher-order accurate T.V.D. schemes: as an example we would like to mention the monotone upwind schemes for conservation laws (M.U.S.C.L.) introduced by van Leer.

Here two remarks ought to be made: one is that no theoretical result of either T.V.D. (w.r.t. time) or monotony of the weak solution was established for nonlinear multidimensional systems (like the Euler equations) which are nevertheless the principal application in practice. In fact the minimal requirement there, is preserving positiveness. Secondly, while the T.V.D. concept is clearly and uniquely defined in the one-dimensional case, it will be open to discussion in a higher dimensional space. This might explain the flowering of numerous T.V.D. or T.V.D.-like schemes and methods which are often well adapted to the only one special case (maybe a class) that was the basis of their construction and experimentation; for instance a scheme well known for its nice properties when applied to transonic flows may well fail for larger Mach numbers or when new terms are added to the equations (viscous terms) or in presence of nonlinear source terms (reactive flows). This is because the main properties of the equations may change from one application to another (loss of hyperbolicity for instance).

In this paper, we concentrate on the multidimensional case, studied with unstructured meshes. Section 1 of the present paper is devoted to the scalar linear case in one and two space dimensions. Extensions of the first-order accurate upwind scheme to the second-order one are presented. A particular attention is given to the change of space dimension. A monotony preserving upwind scheme is introduced in the two-dimensional finite-element context. The application to the 2-D Euler equations of the M.U.S.C.L. with different limitation procedures are presented. Some numerical experiments are shown.

Among the difficulties one can meet and beside the classical stability problems are firstly, preserving positiveness for some physically significant variables, like pressure, density, temperature, species concentration and secondly, avoiding spurious oscillations.

The basic choices we made are: (1) the use of triangular (tetrahedral) meshes for geometric flexibility, (2) finite-volume formulation for conservativeness and (3) upwinding in order to approximate well the convective terms of the equations. All these ingredients will result in a finite-volume/finite-element conservative upwind scheme.

The basic upwind scheme is known to be monotone in the case of a one- dimensional scalar equation, but then, it is only first-order accurate. A second class of schemes were then derived from the ideas of preserving monotony and of stability conditions for the nonlinear scalar equation. These schemes are known as total variation diminishing schemes (T.V.D.). The most interesting T.V.D. schemes are the nonlinear ones, as linearity would imply monotony and thus first-order accuracy.

This led to constructing new schemes or more often developing techniques for transforming existing schemes (that is, first-order upwind schemes, centered schemes, F.E.M Ritz-Galerkin...) into higher-order accurate T.V.D. schemes: as an example we would like to mention the monotone upwind schemes for conservation laws (M.U.S.C.L.) introduced by van Leer.

Here two remarks ought to be made: one is that no theoretical result of either T.V.D. (w.r.t. time) or monotony of the weak solution was established for nonlinear multidimensional systems (like the Euler equations) which are nevertheless the principal application in practice. In fact the minimal requirement there, is preserving positiveness. Secondly, while the T.V.D. concept is clearly and uniquely defined in the one-dimensional case, it will be open to discussion in a higher dimensional space. This might explain the flowering of numerous T.V.D. or T.V.D.-like schemes and methods which are often well adapted to the only one special case (maybe a class) that was the basis of their construction and experimentation; for instance a scheme well known for its nice properties when applied to transonic flows may well fail for larger Mach numbers or when new terms are added to the equations (viscous terms) or in presence of nonlinear source terms (reactive flows). This is because the main properties of the equations may change from one application to another (loss of hyperbolicity for instance).

In this paper, we concentrate on the multidimensional case, studied with unstructured meshes. Section 1 of the present paper is devoted to the scalar linear case in one and two space dimensions. Extensions of the first-order accurate upwind scheme to the second-order one are presented. A particular attention is given to the change of space dimension. A monotony preserving upwind scheme is introduced in the two-dimensional finite-element context. The application to the 2-D Euler equations of the M.U.S.C.L. with different limitation procedures are presented. Some numerical experiments are shown.

##### MSC:

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |