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

Efficient high-order discontinuous Galerkin schemes with first-order hyperbolic advection–diffusion system approach

“…These results are particularly remarkable compared with the highorder RD schemes proposed in Refs. [7,9], which despite its attractiveness and ability to predict discontinuous solutions accurately, predicted less accurate discontinuous solution gradients.…”

“…We will define the constructions of C k , V k , and their relations to the original basis functions, B k , and the vector of unknown polynomial coefficients, u k , which are given in Eq. (9), shortly in this section. At the end of this process, we apply the Galekrin discretization by multiplying the hyperbolic advection-diffusion system by the modified basis functions, C k , and perform an integration by part to arrive at…”

“…The schemes that are designed with the first-order hyperbolic advection-diffusion system have shown [4,5,6,7,8] to produce the same high-order solutions for both the primal variables and their gradients on irregular grids. We have also shown [7,9] that one can design high-order schemes to obtain oscillation-free solution gradients on arbitrary triangular elements. The price to pay for these high-order schemes is the increase of the degrees-of-freedom (DoF) compared to the conventional schemes, that are designed based on the target governing equations.…”

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

“…The resulting scheme is therefore multidimensional upwind, positive, and linearity preserving. Many variants can be constructed depending on the choice of . Here, we will use the shock detector function of Dobeš and Deconinck; this combination is known in the literature as the B x scheme.…”

“…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.…”