Raaghav Ramani

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].

Raaghav Ramani

We propose a time-domain boundary integral method to model linear wave propagation with refractive, focusing, and Doppler effects arising from medium heterogeneities and moving obstacles. In contrast to existing techniques, our method avoids volumetric discretization and yields a formulation posed only on the boundary of the obstacle. We combine two classical ingredients: a geometric--optics parametrix to capture leading-order behavior at propagating wavefronts, and a ray-based characterization of the distorted causal geometry. The former provides a framework for defining layer potentials and deriving the associated boundary integral equations, while the latter enables a pure boundary-only formulation. Taken together, these ingredients extend existing numerical techniques for the homogeneous, fixed-boundary case to the heterogeneous, moving-boundary setting, with appropriate modifications to capture the discrete causal structure arising from the intersection of distorted light cones with the worldsheet of the moving boundary. Numerical experiments demonstrate the ability of the method to resolve Doppler effects from moving obstacles, including a rotating turbine configuration, with stable performance up to Mach 0.9. In heterogeneous media, the method captures strong refractive effects from spherical inclusions: wave propagation wrapping around a gas bubble in water, and defocusing from a hot fireball rising through a stratified atmosphere.