Institute for Computational and Experimental Research in Mathematics (ICERM)

Abstract

Second order phase transitions describe in a macroscopic manner the decay of energy of a physical system Applied to fluid turbulence, they lead to a proof of existence of non-smooth solutions for the Navier-Stokes equation, with the mathematical proof outlined here. Applied to Yang-Mills quantum fields, they give rise to the mass gap, a Millenium Prize problem. Applied to cosmology and astrophysics, they support a novel explanation for dark matter and dark energy.

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Abstract

We will discuss current work on fourth-order accurate finite-volume methods for conservation laws on locally-rectangular adaptive grids, including discretization methods for mapped-multiblock coordinate systems, and the use of embedded domain-specific languages and run-time compilation for obtaining high performance and high productivity on CPU and GPU-based systems. We will illustrate the use of these methods for CFD calculations in spherical geometries.

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When solving partial differential equations, finite difference methods have the advantage of simplicity, however they are usually only designed on Cartesian meshes. In this talk, we will discuss a class of high order finite difference numerical boundary condition for solving hyperbolic Hamilton-Jacobi equations, hyperbolic conservation laws, and convection-diffusion equations on complex geometry using a Cartesian mesh. The challenge results

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Fully resolved simulations of multiphase flow have come a long way in the last two to three decades and we now routinely simulate flows where we fully resolve a large range of temporal and spatial continuum scales, at least for relatively simple disperse flows of bubbles, drops and solid particles. The challenges now are twofold: How to use the results to increase our ability to predict industrial scale flows and how to conduct direct numerical simulations of much more complex systems. Routine predictions usually require course models where the large scales are evolved deterministically, and small scales are included statistically. We will discuss strategies to coarsen results of fully resolved simulations of multiphase flows in a systematic way, retaining sharp interfaces, as well as initial efforts to develop models to evolve the coarse flow. We also discuss efforts to simulate more complex flows, including three-phase liquid-gas-solid disperse flows where the solid particles are either hydrophobic or hydrophilic.

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Numerical software is being reinvented to provide opportunities to tune dynamically the accuracy of computation to the requirements of the application, resulting in savings of memory, time, and energy. Floating point computation in science and engineering has a history of “oversolving” relative to expectations for many models. So often are real datatypes defaulted to double precision that GPUs did not gain wide acceptance until they provided in hardware operations not required in their original domain of graphics. However, computational science is now reverting to employ lower precision arithmetic where possible. Many matrix operations considered at a blockwise level allow for lower precision and many blocks can be approximated with low rank near equivalents. This leads to smaller memory footprint, which implies higher residency on memory hierarchies, leading in turn to less time and energy spent on data copying, which may even dwarf the savings from fewer and cheaper flops. We provide examples from several application domains, including a look at campaigns in geospatial statistics, seismic processing, genome wide association studies, and climate emulation that earned Gordon Bell Prize finalist status in 2022, 2023, and 2024.

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We consider a general system of hyperbolic-parabolic balance laws, which contains both second order dissipation (viscosity) and lower order damping. As dictated by physics, the viscosity matrix and the Jacobian matrix of the lower order term are usually rank deficient. We study the Green's function of its linearization around a constant equilibrium state. The Green's function is constructed using heat kernels along equilibrium characteristics directions and delta functions along frozen characteristics direction. A detailed error estimate is given in space-time pointwise sense. Our result applies to polyatomic gas flows.

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One of the three modes of heat transport is convection. This detail matters in many contexts: interior of the sun, interior as well as the atmosphere of the earth, nuclear cooling structures, and many engineering applications. For example, an improper understanding of convection inside the earth led Lord Kelvin to underestimate the age of the earth (as is well known) by a factor of a few hundreds. We discuss turbulent convection in this talk. In particular, we will examine its behavior at parameter values characteristic of geophysical and astrophysical problems, taking guidance from experiment, simulation, and theory.

Abstract

Interface and mixing and their non-equilibrium kinetics and dynamics couple micro to macro scales, and are ubiquitous to occur in fluids, plasmas and materials, in high energy density regimes. Stellar evolution, plasma fusion, reactive fluids, purification of water, and nanofabrication are a few examples of many processes to which dynamics of interfaces is directly relevant. This talk presents the rigorous theory of the stability of the interface – a phase boundary broadly defined. We directly link the structure of macroscopic flow fields to microscopic interfacial transport, quantify the contributions of macro and micro stabilization mechanisms to interface stability, and discover the fluid instabilities never previously discussed. In ideal and realistic fluids, the interface stability is set primarily by the interplay of the macroscopic inertial mechanism balancing the destabilizing acceleration, whereas microscopic thermodynamics create vortical fields in the bulk. By linking micro to macro scales, the interface is the place where balances are achieved.

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Abstract

We will discuss some recent progress of our research program in developing rigorous mathematical approaches, techniques, and ideas to analyze the nonlinear stability/instability of characteristic interfaces (including vortex sheets and entropy waves) and related free boundary problems for compressible fluid flows governed by the Euler equations and related nonlinear PDEs, and explore stabilizing mechanisms such as magnetic, relativistic, thermoelastic, and compressibility effects for the characteristic interfaces. We also discuss how the principle of maximum entropy provides an approach for determining the physical probability distributions of the turbulent flow in the presence of constraints on the mean energy, if time permits.