Steve Shkoller

Jon Reisner, Jonathan Serencsa, Steve Shkoller

We introduce a new methodology for adding localized, space-time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the $C$-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction-diffusion equation to our system of conservation laws, whose solution $C(x,t)$ is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, $C(x,t)$ is large and localized, and transitions smoothly in space-time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. We demonstrate the effectiveness of the $C$-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our $C$-method framework, WENO-C. Third, we use WENO with the Lax-Friedrichs flux together with the $C$-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.

C. H. Arthur Cheng, Rafael Granero-Belinchon, Steve Shkoller et al.

We develop a rigorous asymptotic derivation for two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting epsilon denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in epsilon to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water waves system is obtained as an infinite sum of solutions to linear problems at each epsilon^k level, and truncation of this series leads to our two asymptotic models, that we call the quadratic and cubic h-models. Using the growth rate of the Catalan numbers (from number theory), we prove well-posedness of the h-models in spaces of analytic functions, and prove error bounds for solutions of the h-models compared against solutions of the water waves system. We also show that the Craig-Sulem models of water waves can be obtained from our asymptotic procedure and that their WW2 model is well-posed in our functional framework. We then develop a novel numerical algorithm to solve the quadratic and cubic h-models as well as the full water waves system. For three very different examples, we show that the agreement between the model equations and the water waves solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves.

Raaghav Ramani, Jon Reisner, Steve Shkoller

This is the second part to our companion paper [18]. Herein, we generalize to two space dimensions the C-method developed in [20,18] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities. For gas dynamics, the C-method couples the Euler equations to scalar reaction-diffusion equations, which we call C-equations, whose solutions serve as space-time smooth artificial viscosity indicators for shocks and contacts. We develop a high-order numerical algorithm for gas dynamics in 2-D which can accurately simulate the Rayleigh-Taylor (RT) instability with Kelvin-Helmholtz (KH) roll-up of the contact discontinuity, as well as shock collision and bounce-back. We implement both directionally isotropic and anisotropic artificial viscosity schemes, the latter adding diffusion only in directions tangential to the evolving front. We additionally produce a novel shock collision indicator function, which naturally activates during shock collision, and then smoothly deactivates. Moreover, we implement a high-frequency 2-D wavelet-based noise detector together with an efficient and localized noise removal algorithm. We provide numerical results for some classical 2-D test problems, including the RT problem, the Noh problem, a circular explosion problem from the Liska and Wendroff [13] review paper, the Sedov blast wave problem, the double Mach 10 reflection test, and a shock-wall collision problem. In particular, we show that our artificial viscosity method can eliminate the wall-heating phenomenon for the Noh problem, and thereby produce an accurate, non-oscillatory solution, even though our simplified WENO-type scheme fails to run for this problem.

Raaghav Ramani, Jon Reisner, Steve Shkoller

In this first part of two papers, we extend the C-method developed in [40] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities in one space dimension. For gas dynamics, the C-method couples the Euler equations to a scalar reaction-diffusion equation, whose solution $C$ serves as a space-time smooth artificial viscosity indicator. The purpose of this paper is the development of a high-order numerical algorithm for shock-wall collision and bounce-back. Specifically, we generalize the original C-method by adding a new collision indicator, which naturally activates during shock-wall collision. Additionally, we implement a new high-frequency wavelet-based noise detector together with an efficient and localized noise removal algorithm. To test the methodology, we use a highly simplified WENO-based discretization scheme. We show that our scheme improves the order of accuracy of our WENO algorithm, handles extremely strong discontinuities (ranging up to nine orders of magnitude), allows for shock collision and bounce back, and removes high frequency noise. The causes of the well-known "wall heating" phenomenon are discussed, and we demonstrate that this particular pathology can be effectively treated in the framework of the C-method. This method is generalized to two space dimensions in the second part of this work [41].