NIST-09 (Wave Front)¶

Commonly used for testing adaptive refinement algorithms is Poisson’s equation with a solution that has a steep wave front in the interior of the domain.

Model problem¶

Equation solved: Poisson equation

(1) $-\Delta u - f = 0.$

Domain of interest: Unit Square $(0, 1)^2$

Boundary conditions: Dirichlet, given by exact solution.

Exact solution¶

$u(x, y) = tan^{-1}(\alpha (r - r_{0}))$

where $r = \sqrt{(x - x_{loc})^{2} + (y - y_{loc})^{2}}$ , $(x_{loc}, y_{loc})$ is the center of the circular wave front, $r_{0}$ is the distance from the wave front to the center of the circle, and $\alpha$ gives the steepness of the wave front.

Material parameters¶

This benchmark has four different versions, we use the global variable PARAM (below) to switch among them.

int PARAM = 3;     // PARAM determines which parameter values you wish to use
                   // for the steepness and location of the wave front.
                   //  | name       | ALPHA | X_LOC | Y_LOC | R_ZERO
                   // 0: mild         20      -0.05   -0.05    0.7
                   // 1: steep        1000    -0.05   -0.05    0.7
                   // 2: asymmetric   1000     1.5     0.25    0.92
                   // 3: well         50       0.5     0.5     0.25