## Spectral Difference Method

The SD method is an efficient high-order method for solving the Navier-Stokes equations. We take the two-dimensional Navier-Stokes equations as an example to explain how it works. The equations can be written in the following form: $$\frac{\partial \mathbf{Q}} {\partial t} + \frac{\partial \mathbf{F}} {\partial x} + \frac{\partial \mathbf{G}} {\partial y} = 0,$$ where $\mathbf{Q}=[\rho,~ \rho u, ~\rho v,~\rho E]^\mathsf{T}$ is the vector of conservative variables, $\mathbf{F}=\mathbf{F}_{inv}+\mathbf{F}_{vis}$ and $\mathbf{G}=\mathbf{G}_{inv}+\mathbf{G}_{vis}$ are the flux vectors. The subscripts '$inv$' and '$vis$' represent inviscid and viscous fluxes whose expressions are $$\mathbf{F}_{inv} = \left[ \begin{array}{c} \rho u \\[0.3em] \rho u^2 + p \\[0.3em] \rho uv \\[0.3em] (\rho E+p)u \end {array} \right], ~ \mathbf{F}_{vis} =-\left[ \begin{array}{c} 0 \\[0.3em] \tau_{xx} \\[0.3em] \tau_{yx} \\[0.3em] u\tau_{xx}+v\tau_{yx}-q_x \end{array} \right], ~ \mathbf{G}_{inv} = \left[ \begin{array}{c} \rho v \\[0.3em] \rho uv \\[0.3em] \rho v^2 + p \\[0.3em] (\rho E+p)v \end {array} \right], ~ \mathbf{G}_{vis} =-\left[ \begin{array}{c} 0 \\[0.3em] \tau_{xy} \\[0.3em] \tau_{yy} \\[0.3em] u\tau_{xy}+v\tau_{yy}-q_y \end{array} \right].$$

In SD method on quadrilateral grids, each grid element is mapped from the physical space $(x,y)$ to a standard unit square element in the computational space $(\xi,\eta)$ using, for example, iso-parametric mapping. The mapping also transforms the Navier-Stokes equations into the following conservative form in the computational space: $$\frac{\partial \widetilde{\mathbf{Q}}} {\partial t} + \frac{\partial \widetilde{\mathbf{F}}} {\partial \xi} + \frac{\partial \widetilde{\mathbf{G}}} {\partial \eta} = 0,$$ where $\widetilde{\mathbf{Q}}$, $\widetilde{\mathbf{F}}$ and $\widetilde{\mathbf{G}}$ are the vectors of computational variable and fluxes that are related to the physical ones as $$\widetilde{\mathbf{Q}} = |\mathcal{J}|\mathbf{Q}, ~~ \left( \begin{array}{c} \widetilde{\mathbf{F}} \\[0.3em] \widetilde{\mathbf{G}} \end{array} \right) = |\mathcal{J}|\mathcal{J}^{-1} \left( \begin{array}{c} \mathbf{F} \\[0.3em] \mathbf{G} \end{array} \right),$$ where $\mathcal{J}=\partial (x,y)/\partial (\xi,\eta)$ is the Jacobian matrix for mapping, $|\mathcal{J}|$ is its determinant, and $\mathcal{J}^{-1}$ is its inverse. Solution points (SPs) $X_s$ and flux points (FPs) $X_{s+1/2}$ are then defined within each standard element to facilitate the construction of solution and flux polynomials. The figure below shows the distribution of SPs and FPs for a third-order scheme, where red circles denote SPs and blue squares denote FPs. With that, the following Lagrange interpolation bases are readily available at each SP and FP for a standard element, $$h_i(X) = \prod_{s=1,s\neq i}^{N}\left(\frac{X-X_s}{X_i-X_s}\right), ~~ l_{i+1/2}(X) = \prod_{s=0,s\neq i}^{N}\left(\frac{X-X_{s+1/2}}{X_{i+1/2}-X_{s+1/2}}\right).$$ They are employed to construct the solution and flux polynomials using tensor product forms as, \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), \\[0.5em] \widetilde{\mathbf{F}}(\xi,\eta) &= \sum_{j=1}^{N} \sum_{i=0}^{N} \widetilde{\mathbf{F}}_{i+1/2,j} l_{i+1/2}(\xi) h_j(\eta), \\[0.5em] \widetilde{\mathbf{G}}(\xi,\eta) &= \sum_{j=0}^{N} \sum_{i=1}^{N} \widetilde{\mathbf{G}}_{i,j+1/2} h_i(\xi) l_{j+1/2}(\eta). \end{aligned} The above polynomials are only element-wise continuous, but discontinuous across cell boundaries. Therefore, common solution and common fluxes need to be defined on cell boundaries. They are computed, for instance, by averaging and using Riemann solver, etc. Once the common values are obtained, one can compute the residuals by direct differentiation of the continuous polynomials and update the solutions using either explicit or implicit time-marching schemes.

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