waveDG

This library implements the Discontinuous Galerkin Finite Element Method in 2D with emphasis on application to the linear wave equation as a first order system (e.g. linearized Euler equations): $$p_t + \nabla \cdot \mathbf{u} = 0,$$ $$\mathbf{u}_t + \nabla p = 0.$$

The code currently supports Dirichlet, Neumann, and (approximate) outflow boundary conditions.

DG Formulation

For the linear hyperbolic conservation law: $$\mathbf{q}_t + \partial_x (A \mathbf{q}) + \partial_y (B \mathbf{q}) = 0.$$ Let $I$ be an element on the mesh. We define the approximation space $V_h\subset L^2(I)$ to be the space of polynomials upto degree $P$ on $I$. The DG weak form of the equation on each element $I$ is $$(\mathbf{q}_t, v)_I - (A\mathbf{q}, v_x)_I - (B \mathbf{q}, v_y)_I + \langle \mathbf{q}^\star, [v] \rangle_{\partial I} = 0, \qquad \forall v\in V_h.$$ Where $$\mathbf{q}^\star = a C\{\mathbf{q}\} + b |C|\mathbf[\mathbf{q}], \quad C = n_x A + n_y B.$$ Here $(n_x, n_y)$ is the normal to $\partial I$, and if $C = R\Lambda R^{-1}$ is the eigenvalue decomposition of $C$, then $|C| = R|\Lambda|R^{-1}$ where $|\Lambda|$ has the absolute values of the eigenvalues of $C$ on the diagonal. Note that it is necessary that $C$ have real eigenvalues for any $(n_x, n_y)$, which is also a condition for the well-posedness of the PDE. We can choose $a = 1, b = 1/2$ for the upwind scheme, or $a = 1, b = 0$ for the central (average) scheme.

Compiling the Library

To compile the serial version of the library run:

cmake .
make wavedg -j

To compile the MPI version of the library run:

cmake . -D WDG_USE_MPI=ON
make wavedg -j

In either cases, you can add the flag -D WDG_DEBUG=ON to enable debugging (-g) and add additional correctness checks.

To install the library run sudo make install.