Contributed Talk
Benefits in Predicting Future Values in Implicit Time Discretizations
Slovak University of Technology, Slovakia
Abstract
In this work, we address implicit numerical schemes for solving level set equations
where the use of large time steps is desired, particularly in scenarios involving
velocity fields with a large variation in space or numerical solutions approaching
stationary states.
Although level set methods are typically applied to continuous functions, their gradients
can develop discontinuities. We focus on stable numerical methods that reduce non-physical
oscillations in the solution's gradient.
To address these issues, we propose a novel finite difference scheme based on a predictor-corrector
methodology [3], with a fixed finite number of corrector steps.
The scheme leverages predicted future points in time [5] and space,
thereby allowing oscillation reduction regardless of the Courant number through limiting
opposite to other approaches [2].
The method achieves up to third-order accuracy in both space and time.
An efficient fast sweeping solver [4] is used to solve the system.
The results and properties of the proposed method are supported by examples of two-dimensional
advective level set equations.
References
[1]
Zhao, H. (2005). A fast sweeping method for eikonal equations. Math. Comput., 74(250), 603⧿627.
[2]
Puppo, G., Semplice, M., & Visconti, G. (2022). Quinpi: Integrating Conservation Laws with CWENO Implicit Methods. Communications on Applied Mathematics and Computation, 343⧿369.
[3]
Micalizzi, L., Torlo, D., & Boscheri, W. (2025). Efficient iterative arbitrary high-order methods: an adaptive bridge between low and high order. Communications on Applied Mathematics and Computation, 7(1), 40⧿77.
[4]
Luo, S., & Zhao, H. (2016). Convergence analysis of the fast sweeping method for static convex Hamilton⧿Jacobi equations. Research in the Mathematical Sciences, 3(1), 35.
[5]
Cash, J.R. (2000). Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs. Journal of Computational and Applied Mathematics, 125(1-2), 117⧿130.
Acknowledgments:
Funded by the EU NextGenerationEU through the Recovery and Resilience Plan for Slovakia under the project No. 09I03-03-V05-00005.
Joint work with Peter Frolkovič