# Testing the SUPG method and other methods for hyperbolic equations

I am interesting in integrating the simple equation $$\frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla \phi = 0$$ with a Dirichlet boundary condition at the influx boundary ($$\mathbf{u} \cdot \mathbf{n} < 0$$). The domain is a simple unit square. I pretend to use SUPG and I want to validate the method using the method of manufactured solutions. I came up with a simple solution

$$\phi = \begin{cases} cos(5x - t) ~ \text{for} ~ y > 0.5 \\ 0 ~ \text{else} \end{cases}$$

with a advective field $$\mathbf{u} = (1.0/5.0, 0)$$. This makes the source term of the equation equal to zero and I adjust the Dirichlet BC appropriately

Discretizing this equation with just Lagrange elements, no stabilization methods added, and integrating in time with Crank-Nicholson results in nice solutions with no oscillations that I would expect from using the Lagrange elements on a hyperbolic equation with a discontinuous function being advected. Why is it so? Are there other manufactured solutions that will actually trigger oscillations?

• Isn't that the whole idea of SUPG, to add upwind stabilization to reduce oscillations? – Vikram Jun 9 at 13:59
• An harder test case would be to advect a step function as in a Cauchy-Riemann problem. There you would get oscillations without SUPG. – BlaB Jun 13 at 14:00