Using About Auxiliary Equation-Based Nodes it is possible to solve one or several first-order wave equations; that is, PDEs of the form
where u is the unknown,
da the mass coefficient,
f the source, and
Γ the flux vector, which generally depends on
u.
A basis for V is given by the nodal discontinuous Lagrange shape functions
shhwdisc (see
Discontinuous Lagrange (shdisc) and Nodal Discontinuous Lagrange Elements (shhwdisc) in the
COMSOL Multiphysics Programming Reference Manual. This basis is tailor made for this type of discontinuous Galerkin method, and it has nearly optimal interpolation properties.
The next step is to integrate by parts. Some care must be taken, since the integrands v and
u are discontinuous functions across element boundaries and only continuous in the interior of each element. Therefore, the integrals are first written as a sum over the elements and then integration by parts is done on each element, which gives
where n is the outward unit normal on the element. Further,
is the so-called numerical flux, which defines the flux vector on each element boundary. The flux vector is usually discontinuous because it depends on
u.
The numerical flux defines how adjacent elements are connected and how continuous u is. Different definitions of the numerical flux lead to different variants of discontinuous Galerkin methods.
where u_ is the inside values and
u+ is the outside value. So it is the element’s value minus the other element’s value. The multiplication is an outer product in the case of a vector quantity
u.
where p is the order of the shape functions and
C a generic constant, typically 0.25.
As implemented in COMSOL Multiphysics, the nodal discontinuous Lagrange shape functions are the only set of shape functions defined for this interface. The associated element order can be chosen from the
Element order list. The highest available order is four, and the default order is two.