In this talk, we will discuss some recent results that widen the class of elliptic partial differential equations (PDEs) with provably convergent schemes, and discuss some techniques to improve their accuracy.
The approximation theory of Barles and Souganidis guarantees the convergence of numerical schemes to the unique viscosity solution of the equation provided they are monotone, stable, and consistent. However, in general, these schemes are only first-order accurate for first-order equations and second-order accurate for second-order equations.
We will begin with first-order Hamilton-Jacobi equations, with a case study of the Eikonal equation, and introduce the notion of filtered schemes. The idea is to blend a stable and monotone convergent scheme with an accurate scheme and retain the advantages of both: stability and convergence of the former, and higher accuracy of the latter. In our case, by judiciously choosing which schemes to use, as well as how to blend them, we are able to construct explicit schemes that allow the use of the efficient fast sweeping method.
In the second part of this talk, we will discuss some of the challenges in building monotone schemes. To do this, we will consider two equations: (i) the 2-Hessian equation and (ii) the nonlinear partial differential equation that governs the motion of level sets by affine curvature. Speaker(s): Tiago Salvador (University of Michigan)
The approximation theory of Barles and Souganidis guarantees the convergence of numerical schemes to the unique viscosity solution of the equation provided they are monotone, stable, and consistent. However, in general, these schemes are only first-order accurate for first-order equations and second-order accurate for second-order equations.
We will begin with first-order Hamilton-Jacobi equations, with a case study of the Eikonal equation, and introduce the notion of filtered schemes. The idea is to blend a stable and monotone convergent scheme with an accurate scheme and retain the advantages of both: stability and convergence of the former, and higher accuracy of the latter. In our case, by judiciously choosing which schemes to use, as well as how to blend them, we are able to construct explicit schemes that allow the use of the efficient fast sweeping method.
In the second part of this talk, we will discuss some of the challenges in building monotone schemes. To do this, we will consider two equations: (i) the 2-Hessian equation and (ii) the nonlinear partial differential equation that governs the motion of level sets by affine curvature. Speaker(s): Tiago Salvador (University of Michigan)
| Building: | East Hall |
|---|---|
| Event Type: | Workshop / Seminar |
| Tags: | Mathematics |
| Source: | Happening @ Michigan from Department of Mathematics, Applied Interdisciplinary Mathematics (AIM) Seminar - Department of Mathematics |
