Illustrating the placement of the field components. For heterogeneous media then is placed together with and together with the corresponding velocity components.
A staircase approximation of a rigid boundary not aligned with the grid. A cross showing the stencil for the cell centered at is shown. The point which we Taylor expand around can be chosen anywhere along inside the cell. See Fig. 3. for the other 7 cases the boundary can cut the update stencil for the pressure.
The eight different cases for how the boundary can intersect the update stencil for in each coordinate plane. The cross represents the computational molecule with in the center, and the extra line is the boundary. See Fig. 2 for a specific case (NE).
Illustrating the discretization of soft boundaries. Circled velocity points have modified coefficients corresponding to interpolation and dashed circles indicate coefficients giving extrapolation.
Illustrating the angles and . The normal is pointing into the domain.
Convergence result in and for test case 1 (flat surface). A line indicating slope 1 is included. In we clearly see the behavior of standard staircasing, while when modifying the stencils we obtain . In then standard staircasing exhibit first order convergence for coarse grids, while eventually settling on , showing that the errors generated at the boundary slowly decay.
Convergence result in and for test case 2 (sphere). The setup for this test case is illustrated in Fig. 8. A line indicating slope 1 is included. Again a clear improvement in convergence rate is observed, especially in , where the inconsistencies are removed.
(Color online) Comparing the computed field in test case 2 (sphere) for a slice in at the end time . The large errors generated at the surface of the sphere for standard staircasing is clearly seen. After the coefficients are modified the quality of the solution improves significantly. Note that while we compute in the domain , we only measure the error in to isolate the errors generated by the sphere. The convergence for this setup is shown in Fig. 7.
The error in the computed pressure and velocity field for soft boundaries, measured in and . Solid lines indicating slope 1 and 2 are included. This is the test case defined by (20). Compared to rigid boundaries the solution show much higher regularity, making the improvements with modified coefficients more clear. For both pressure and velocity and in and the convergence rate is improved by one. We see that it is actually only in the velocity components that standard staircasing is inconsistent.
The norm as a function of the number of timesteps for two different CFL numbers just above and below the free space limit. The field is initialized to random data to excite all frequencies representable on the grid. (top) Rigid boundary (test case 2). (bottom) Soft boundary.
Consistency conditions for the plane. The angle is given with respect to the axis, i.e., corresponds to a vertical boundary.
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