Complex Geometries

Sandra May works at the Seminar for Applied Mathematics at ETH
Zurich and visited Karlsruhe for a talk at the CRC Wave
phenomena. Her research is in numerical analysis, more
specifically in numerical methods for solving PDEs. The focus is
on hyperbolic PDEs and systems of conservation laws. She is both
interested in theoretical aspects (such as proving stability of a
certain method) and practical aspects (such as working on
high-performance implementations of algorithms). Sandra May
graduated with a PhD in Mathematics from the Courant Institute of
Mathematical Sciences (part of New York University) under the
supervision of Marsha Berger. She likes to look back on the
multicultural working and learning experience there.


We talked about the numerical treatment of complex geometries.
The main problem is that it is difficult to automatically
generate grids for computations on the computer if the shape of
the boundary is complex. Examples for such problems are the
simulation of airflow around airplanes, trucks or racing cars.
Typically, the approach for these flow simulations is to put the
object in the middle of the grid. Appropriate far-field boundary
conditions take care of the right setting of the finite
computational domain on the outer boundary (which is cut from an
infinite model). Typically in such simulations one is mainly
interested in quantities close to the boundary of the object.


Instead of using an unstructured or body-fitted grid, Sandra May
is using a Cartesian embedded boundary approach for the grid
generation: the object with complex geometry is cut out of a
Cartesian background grid, resulting in so called cut cells where
the grid intersects the object and Cartesian cells otherwise.
This approach is fairly straightforward and fully automatic, even
for very complex geometries. The price to pay comes in shape of
the cut cells which need special treatment. One particular
challenge is that the cut cells can become arbitrarily small
since a priori their size is not bounded from below. Trying to
eliminate cut cells that are too small leads to additional
problems which conflict with the goal of a fully automatic grid
generation in 3d, which is why Sandra May keeps these potentially
very small cells and develops specific strategies instead.


The biggest challenge caused by the small cut cells is the small
cell problem: easy to implement (and therefore standard) explicit
time stepping schemes are only stable if a CFL condition is
satisfied; this condition essentially couples the time step
length to the spatial size of the cell. Therefore, for the very
small cut cells one would need to choose tiny time steps, which
is computationally not feasible. Instead, one would like to
choose a time step appropriate for the Cartesian cells and use
this same time step on cut cells as well.


Sandra May and her co-workers have developed a mixed explicit
implicit scheme for this purpose: to guarantee stability on cut
cells, an implicit time stepping method is used on cut cells.
This idea is similar to the approach of using implicit time
stepping schemes for solving stiff systems of ODEs. As implicit
methods are computationally more expensive than explicit methods,
the implicit scheme is only used where needed (namely on cut
cells and their direct neighbors). In the remaining part of the
grid (the vast majority of the grid cells), a standard explicit
scheme is used. Of course, when using different schemes on
different cells, one needs to think about a suitable way of
coupling them.


The mixed explicit implicit scheme has been developed in the
context of Finite Volume methods. The coupling has been designed
with the goals of mass conservation and stability and is based on
using fluxes to couple the explicit and the implicit scheme. This
way, mass conservation is guaranteed by construction (no mass is
lost). In terms of stability of the scheme, it can be shown that
using a second-order explicit scheme coupled to a first-order
implicit scheme by flux bounding results in a TVD stable method.
Numerical results for coupling a second-order explicit scheme to
a second-order implicit scheme show second-order convergence in
the L^1 norm and between first- and second-order convergence in
the maximum norm along the surface of the object in two and three
dimensions.


We also talked about the general issue of handling shocks in
numerical simulations properly: in general, solutions to
nonlinear hyperbolic systems of conservation laws such as the
Euler equations contain shocks and contact discontinuities, which
in one dimension express themselves as jumps in the solution. For
a second-order finite volume method, typically slopes are
reconstructed on each cell. If one reconstructed these slopes
using e.g. central difference quotients in one dimension close to
shocks, this would result in oscillations and/or unphysical
results (like negative density). To avoid this, so called slope
limiters are typically used. There are two main ingredients to a
good slope limiter (which is applied after an initial polynomial
based on interpolation has been generated): first, the algorithm
(slope limiter) needs to detect whether the solution in this cell
is close to a shock or whether the solution is smooth in the
neighborhood of this cell. If the algorithm thinks that the
solution is close to a shock, the algorithm reacts and adjusts
the reconstruted polynomial appropriately. Otherwise, it sticks
with the polynomial based on interpolation. One commonly used way
in one dimension to identify whether one is close to a shock or
not is to compare the values of a right-sided and a left-sided
difference quotient. If they differ too much the solution is
(probably) not smooth there. Good reliable limiters are really
difficult to find.
Literature and additional material

S. May, M. Berger: An Explicit Implicit Scheme for Cut Cells
in Embedded Boundary Meshes, Preprint available as SAM report,
number 2015-44, 2015.

S. May, M. Berger: A mixed explicit implicit time stepping
scheme for Cartesian embedded boundary meshes, Finite Volumes for
Complex Applications VII - Methods and Theoretical Aspects, pp.
393-400, Springer, 2014.

S. May, M. Berger: Two-dimensional slope limiters for finite
volume schemes on non-coordinate-aligned meshes, SIAM J. Sci.
Comput. 35 (5) pp. A2163-A2187, 2013.