Spring 2025 Schedule
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A Convergent Front Tracking Scheme with an Application to Gas Dynamics
We present a modified Front Tracking (mFT) scheme for hyperbolic systems of conservation laws in one space dimension, in which we allow arbitrarily large nonlinear waves. We build the scheme by introducing and solving a ``generalized Riemann Problem'', which yields exact solutions for finite times. After construction of the scheme, under reasonable assumptions, we show that the mFT approximations converge to a weak* solution of the system. This essentially reduces existence of solutions with large amplitude data to obtaining uniform bounds on the total variation of the approximations. We apply the modified Front Tracking (mFT) scheme to the equations of gas dynamics. We solve the generalized Riemann Problem and define the scheme for both $3\times 3$ and $2\times 2$ systems and show the equivalence of the Eulerian and Lagrangian frames. For the $p$-system, modeling isentropic gas dynamics in a Lagrangian frame, we show that there is no finite accumulation of interaction times, if there is a nonincreasing potential on the approximate solutions constructed through our scheme.
Please note change in usual day and time!
1:00 pm, 306 Snow Hall
Uniqueness and Stability of Riemann Shocks to the full Euler system
In this talk, we will discuss the stability of a Riemann shock solution to the compressible Euler system, which is a self-similar entropy shock connecting two different constant states, in a physical class of vanishing viscosity limits. We focus on the 1D full Euler system and consider the Brenner-Navier-Stokes-Fourier system, an amendment of the Navier-Stokes-Fourier system, to describe the physical perturbation class. The proof is based on the method of a-contraction with shifts. This is a joint work with Moon-Jin Kang (KAIST) and Saehoon Eo (Stanford University).
3:00 pm, 306 Snow Hall
Linearized KdV on the line with a metric graph defect
We study the small amplitude linearization of the Korteweg de Vries equation on the line, but with a defect at $x=0$ represented by a network of finite intervals adjoined at that point, scattering waves. For a representative collection of examples, we obtain explicit contour integral representations of the solution and obtain existence and unicity results for piecewise smooth data. We also discuss extensions to more complex metric graph domains and introduce a serial version of the unified transform method which may be more efficient for such problems.
3:00 pm, 306 Snow Hall
A numerical Riemann-Hilbert approach to the computation of transform pairs
This research presents a unified methodology integrating spectral theory, Riemann-Hilbert problems, and inverse scattering theory to efficiently derive, and numerically implement, transform pairs associated to time-evolution variable-coefficient partial differential equations (PDEs). More specifically, the approach combines analytical formulae with numerical ODE and Riemann-Hilbert methods to efficiently evaluate the forward and inverse transforms, giving a hybrid analytical-numerical method for such PDEs. The method is demonstrated on transforms arising in the solution of the time-dependent SchrÓ§dinger equation, producing an accurate and stable time evolution method that does not require time stepping.
3:00 pm, Online via Zoom