Flux Reconstruction Method

The FR method further improves the efficiency of the SD method by collocating the SPs and FPs in the interior of a computational element. The figure below sketches the distribution of the SPs (red dots) and FPs (bondary blue dots) for a third-order FR method. spfp_fr Similar to the SD method, the solution and flux polynomials are constructed using tensor products of the interpolation bases, Q~(ξ,η)=j=1Ni=1NQ~i,jhi(ξ)hj(η),F~(ξ,η)=j=1Ni=1NF~i,jhi(ξ)hj(η),G~(ξ,η)=j=1Ni=1NG~i,jhi(ξ)hj(η). \begin{aligned} \widetilde{\mathbf{Q}}(\xi,\eta) &= \sum_{j=1}^{N} \sum_{i=1}^{N} \widetilde{\mathbf{Q}}_{i,j} h_i(\xi) h_j(\eta), \\ \widetilde{\mathbf{F}}(\xi,\eta) &= \sum_{j=1}^{N} \sum_{i=1}^{N} \widetilde{\mathbf{F}}_{i,j} h_i(\xi) h_j(\eta), \\ \widetilde{\mathbf{G}}(\xi,\eta) &= \sum_{j=1}^{N} \sum_{i=1}^{N} \widetilde{\mathbf{G}}_{i,j} h_i(\xi) h_j(\eta). \end{aligned} Due to the first-order spatial derivatives on the flux terms in the Navier-Stokes equations, the flux polynomials need to be at least one-degree higher than the solution polynomials to give the correct orders of accuracy. To achieve this goal, the above flux polynomials are reconstructed using higher-degree correction polynomials. For example, the reconstructed (corrected) flux polynomial in the ξ\xi-direction is, F^(ξ)=F~(ξ)+(F~LcomF~L)gL(ξ)+(F~RcomF~R)gR(ξ), \widehat{\mathbf{F}}(\xi) = \widetilde{\mathbf{F}}(\xi) + (\widetilde{\mathbf{F}}_L^{com} - \widetilde{\mathbf{F}}_L) \cdot g_L(\xi) + (\widetilde{\mathbf{F}}_R^{com} - \widetilde{\mathbf{F}}_R) \cdot g_R(\xi), where F~\widetilde{\mathbf{F}} is the original flux, F~Lcom\widetilde{\mathbf{F}}_L^{com} and F~Rcom\widetilde{\mathbf{F}}_R^{com} are the common fluxes on the left and the right boundaries, F~L\widetilde{\mathbf{F}}_L and F~R\widetilde{\mathbf{F}}_R are the original discontinuous boundary fluxes, gLg_L and gRg_R are the left and the right correction polynomials. The correction functions must satisfy the following conditions to ensure continuity, gL(0)=1,gL(1)=0;gR(0)=0,gR(1)=1. g_L(0) = 1, \quad g_L(1) = 0; \quad g_R(0) = 0, \quad g_R(1) = 1. The user can define extra conditions to construct a correction polynomial. Once the fluxes are reconstructed, the remaining steps are essentially identical to those of the SD method.

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