Numerical Formulation

For simplicity we first consider an convection equation for a state vector ${\bf v}$.

$$\diff{{\bf v}}{t} + \diff{{\bf F}^x}{x} + \diff{{\bf F}^y}{y} + \diff{{\bf F}^z}{z} = 0$$ (6)

where ${\bf F}(v)$ is a function of the state vector.

In the discontinuous Galerkin formulation we consider an approximation space $\trsp$ which may contain discontinuous functions. The discrete $\trsp^\delta$ contains polynomials within each ``element'' but zero outside the element. Here the element may be any of the elements we have described and we denote this element by E_i. Thus, the computational domain $\Omega = \bigcup_i E_i,$ and E_i, E_j overlap only on edges. Consequently, each element is treated separately, corresponding to the following variational statement:  

$$\diff{(w,{\bf v})_{E_i}}{t} = -(w,\diff{{\bf F}^x}{x}+\diff{... ...F}^y-{\bf F}^y)n_y+(\hat{\bf F}^z-{\bf F}^z)n_z)_{\partial E_i}$$ (7)

where $w \in S^\delta$.

Computations on each element are performed separately, and the connection between elements is a result of the way boundary conditions are applied. Here, boundary conditions are enforced via the flux ${\bf F}({\bf v})$ that appears in equation (7.2). Because this value is computed at the boundary between adjacent elements, it may be computed from the value of ${\bf v}$ given at either element. These two possible values are denoted here as ${\bf v}^I$ (internal) and ${\bf v}^E$ (external), and the boundary flux written $\hat{{\bf F}}({\bf v}^I, {\bf v}^E)$. Upwinding considerations dictate how this flux is computed. For the case of a hyperbolic system of equations, an approximate Riemann solver would be used to compute a value of $\hat{{\bf F}}$ based on ${\bf v}^I$ and ${\bf v}^E$.