Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier–Stokes equations

Abstract

A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier-Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non-linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution, preserving the optimal accuracy of the method. Linear and non-linear schemes are constructed, and their accuracy is tested with the method of the manufactured solutions. The numerical method is also used for the discretization of smooth and shocked laminar flows in two and three spatial dimensions. Abstract A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier-Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each...

High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics

“…Extensions to other models, such as multiphase flows or Lagrangian hydrodynamics, and further investigations of high order residual distribution schemes will be considered in forthcoming papers. Finally, we are currently extending the proposed approach to viscous problems by combining it with the discretisation technique as explained in [18]. In this case, the time step results, of course, very small, and thus an implicit approach is needed with the challenge to have, nevertheless, a diagonal 'mass matrix'.…”

“…In case of systems, to allow less dissipation, (33) is applied to each variable by considering their characteristic decomposition as described e. g. in [18]. To this end, one considers the eigen-decomposition of the Jacobian matrix A(U ) = ∇ U F(U ) · n of the flux F with respect to the state U K h , where as n we take the average fluid velocity vector or we choose an arbitrary direction (for example the x-coordinate) in case the average velocity vanishes.…”

High-order shock-capturing hyperbolic residual-distribution schemes on irregular triangular grids

“…Imposing a strong BC is not possible for some boundaries and, instead, a weak formulation needs to be constructed. The weak boundary condition is already developed and used within the RD community [7,18,19] . Here, we follow the same procedure that is already employed by the RD community to construct a weak outflow boundary condition, which is the only weak boundary condition used in this study, for hyperbolic adection-diffusion schemes.…”

“…These nonlinear schemes are first developed for the scalar advection equation and later extended for a system of equations [8,9] . These schemes are widely used within the RD community [6,7,[10][11][12] , and applied to advection and advection-diffusion [13][14][15][16] , steady inviscid [12,17,18] , steady Navier-Stokes [19] , turbulent compressible flows [20] , and unsteady [12,21,22] problems. It may also be possible to employ an artificial viscosity technique as widely used in the stabilized finite-element methods, e.g., Refs.…”

An enthalpy‐preserving shock‐capturing term for residual distribution schemes

“…Residual distribution * methods are vertex-centered discretization techniques capable of handling hyperbolic systems of equations on general unstructured simplicial grids. Residual distribution methods were employed successfully for the discretization and solution of advection-diffusion scalar equations [21][22][23][24] and for the compressible Navier-Stokes equations in the context of transonic aerodynamics 19,[25][26][27] in other works. In the hypersonic field, the method has been applied to double-cone configurations 28,29 and blunt-body problems, both in shock-capturing [30][31][32] and shock-fitting [33][34][35][36] contexts.…”