Applied Math Seminar

Date: April 21, 2025

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Todd Arbogast, University of Texas at Austin, Austin

Title: Self Adaptive Theta (SATh) Schemes for Solving Hyperbolic Conservation Laws

Abstract: We present a discontinuity aware quadrature (DAQ) rule, and use it to develop implicit self-adaptive theta (SATh) schemes for the approximation of scalar hyperbolic conservation laws. Our SATh schemes require the solution of a system of two equations, one controlling the cell averages of the solution at the time levels, and the other controlling the space-time averages of the solution. These quantities are used within the DAQ rule to approximate the time integral of the hyperbolic flux function accurately, even when the solution may be discontinuous somewhere over the time interval. We prove that DAQ is accurate to second order when there is a discontinuity in the solution and third order when it is smooth. The resulting scheme is a finite volume theta time stepping method, with theta defined implicitly (or self-adaptively). When upstream weighted, SATh-up is unconditionally stable, satisfies the maximum principle, and is total variation diminishing under appropriate monotonicity and boundary conditions, provided that theta is restricted to be at least 1/2, or even 0 with additional restrictions. We present numerical results showing the performance of the SATh schemes, sometimes using the more general Lax-Friedrichs numerical flux. Compared to solutions of finite volume schemes using Crank-Nicolson and backward Euler time stepping, SATh solutions often approach the accuracy of the former but without oscillation, and they are numerically less diffuse than the later.