# Jet Schemes and Level Set Methods

Information travels along characteristic curves from foot points (blue crosses) to grid points (red dots).

Jet schemes are high-order generalizations of semi-Lagrangian approaches for advection problems. To approximately advect the a given field forward in time, characteristic curves are tracked that go through grid points at the new time step, and the numerical solution is advanced along them. In jet schemes, high order of accuracy is achieved by tracking appropriate derivative information in addition to function values. This derivative data is used to define high-order Hermite p-cubic, p-quintic, etc. interpolants in each grid cell. Through the tracking of derivative data, the high-order accuracy is achieved in an optimally local fashion: the data used to update the solution at a grid point is located only in a single grid cell, independent of the scheme's order.

In the context of level set methods, jet schemes give rise to the gradient-augmented level set method (GALSM). Compared to WENO schemes of the same order and cost, jet schemes provide a better representation of the interface, particularly for thin or slender structures (see subgrid resolution below). In addition, the Hermite interpolant gives rise to high-order approximations to curvature everywhere in the computational domain.

The Hermite tri-cubic interpolant can represent structures of subgrid size, such as a subgrid bubble (left), a subgrid jet (middle), and a subgrid film (right).

Subgrid resolution: A fundamental advantage of using a high-order Hermite interpolant in each each cell is that its zero contour can represent structures of subgrid size. For each of the three subgrid structures shown of the left (bubble, jet, and thin film), the level set function has the same sign at the cell corners. Consequently, a classical level set approach that tracks function values only would not identify the subgrid information; the structure would be lost once it becomes smaller or thinner than the grid resolution. In contrast, the presence of derivative information allows for the representation (and evolution) of subgrid structures.

A Hermite cubic interpolant (red) detecting a subgrid structure in a 1D cell, which the linear interpolant (blue) loses.

The principle why a Hermite cubic interpolant can capture a structure of subgrid size is shown in the figure on the right. Consider a signed distance function (black) in a 1D cell that describes a structure contained within that single cell. Approaches that track function values only would know the two level set values at the grid points, and thus would (unless distance information is explicitly used) not detect a structure. In contrast, if gradients are known at the grid points, then enough data is available to define a Hermite cubic interpolant (red), which, albeit not perfect, does define a subgrid structure. Hence, jet schemes and the GALSM possesses a certain level of subgrid resolution.

Advect-and-project framework: Jet schemes are equivalent to special evolutions in function spaces, namely: evolve a function defined everywhere along all approximate characteristic curves; then, evaluate function values and appropriate derivatives of the advected solution at the grid points, and replace the evolved function by a piecewise Hermite interpolant, defined in each cell via the data at its corner points. This last step is a projection in a function space. Moreover, since only information at grid points influences the function at the end of the step, only characteristic curves that go through grid points at the target time matter. Therefore, jet schemes are equivalent to an advect-and-project framework on functions. This framework in particular allows for proofs of stability (see references below).

## Zalesak Circle and 2D Deformation Field

 Zalesak circle 2D deformation field All computations are conducted on a regular 64 x 64 x 50 grid. In each figure, the gray object denotes the true solution; the red curve is the interface obtained using WENO5 with an SSP RK3 time-stepping scheme; and the blue interface is computed using the third-order GALSM. No re-initialization is used in either case, because the performance of the pure advection scheme is being investigated.   One can observe that the GALSM yields more accurate results than WENO. The potential to capture features of subgrid size is particularly beneficial at preserving the notch in the Zalesak circle, and at not losing the thin structure under the deformation field at maximum deformation.   Video of Zalesak circle Download vid_2d_zalesak.avi (19 MB)    Video of 2D deformation field Download vid_2d_swirl.avi (17 MB) inital conditions and velocity field initial conditions and velocity field Zalesak circle after 1 revolution at maximum deformation Zalesak circle after 4 revolutions at final time

Results for 2D swirl at maximum deformation with jet schemes of 3rd and 5th order.

Jet schemes of higher order: The preceding results demonstrate the accuracy of jet schemes and the GALSM of third order of convergence (i.e., based on bi-cubic interpolation). However, in principle the jet schemes formalism can be extended to arbitrary orders. The figure on the right shows the contours of the 2d deformation field, computed with WENO5 (black), the 3rd order GALSM (blue) and the 5th order GALSM (red). As one can see, the 5th order scheme yields a dramatic improvement over the 3rd scheme: except for a tiny deviation at the slim tail, the true shape (gray) is tracked almost perfectly.
Of course, on the same grid the 5th order jet scheme is computationally more demanding than the 3rd order scheme. In the papers [Discrete Contin. Dyn. Syst. Ser. B 17(4):1229-1259, 2012] and [Int. J. Numer. Anal. Model.-B 3(3):297-306, 2012] a rigorous study of the efficiency of jet schemes (relative to other approaches, such as Discontinuous Galerkin and WENO) in conducted, i.e., which numerical approach yields the highest accuracy at the same computational cost.

## Zalesak Sphere and 3D Deformation Field

 Zalesak sphere 3D deformation field All computations are conducted on a regular 50 x 50 x 50 grid. In each figure, the left object is computed using WENO5 with an SSP RK3 time-stepping scheme; and the right object is computed using the third-order GALSM. No re-initialization is used in either case, because the performance of the pure advection scheme is being investigated.   One can observe that the GALSM yields more accurate results than WENO. The potential to capture features of subgrid size is particularly beneficial at preserving the notch in the Zalesak sphere, and at not losing the thin structure under the deformation field at half time. initial state initial state quarter rotation quarter time half rotation half time three quarter rotation three quarter time one full rotation final time

## Research Support

• 07/2013 - 06/2016, NSF grant DMS-1318709, Collaborative Research: Gradient-augmented level set methods and jet schemes, collaboration with Rodolfo Ruben Rosales
• 07/2008 - 06/2013, NSF grant DMS-0813648, Capturing subgrid structures with level set methods, with Rodolfo Ruben Rosales and Jean-Christophe Nave