The computation of compressibleflows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incom- pressible flow simulation. In this paper, we develop an all-speed asymptotic preserv- ing (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyper- bolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved im- plicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solu- tion techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes. AMS subject classifications: 35Q35, 65M08, 65M99, 76M12, 76N99

/pdf/an-all-speed-asymptotic-preserving-method-for-the-isentropic-64o6rxruoa.pdf

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

Direct numerical simulation of interfacial instabilities: A consistent, conservative, all-speed, sharp-interface method

We present the first three-dimensional, full-star simulations of convection in a white dwarf preceding a Type Ia supernova, specifically the last few hours before ignition. For these long-time calculations, we use our low Mach number hydrodynamics code, MAESTRO, which we have further developed to treat spherical stars centered in a three-dimensional Cartesian geometry. The main change required is a procedure to map the one-dimensional radial base state to and from the Cartesian grid. Our models recover the dipole structure of the flow seen in previous calculations, but our long-time integration shows that the orientation of the dipole changes with time. Furthermore, we show the development of gravity waves in the outer, stable portion of the star. Finally, we evolve several calculations to the point of ignition and discuss the range of ignition radii.

/pdf/low-mach-number-modeling-of-type-ia-supernovae-iv-white-2kn225qdqp.pdf

Low Mach Number Modeling of Type IA Supernovae. IV. White Dwarf Convection

Although climate models have been improving in accuracy and efficiency over the past few decades, it now seems that these incremental improvements may be slowing. As tera/petascale computing becomes massively parallel, our legacy codes are less suitable, and even with the increased resolution that we are now beginning to use, these models cannot represent the multiscale nature of the climate system. This paper argues that it may be time to reconsider the use of adaptive mesh refinement for weather and climate forecasting in order to achieve good scaling and representation of the wide range of spatial scales in the atmosphere and ocean. Furthermore, the challenge of introducing living organisms and human responses into climate system models is only just beginning to be tackled. We do not yet have a clear framework in which to approach the problem, but it is likely to cover such a huge number of different scales and processes that radically different methods may have to be considered. The challenges of multiscale modelling and petascale computing provide an opportunity to consider a fresh approach to numerical modelling of the climate (or Earth) system, which takes advantage of the computational fluid dynamics developments in other fields and brings new perspectives on how to incorporate Earth system processes. This paper reviews some of the current issues in climate (and, by implication, Earth) system modelling, and asks the question whether a new generation of models is needed to tackle these problems.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2008.0207

Developing the next-generation climate system models: challenges and achievements.

Iterated upwind schemes for gas dynamics

https://crd.lbl.gov/assets/pubs_presos/AMCS/ANAG/B114.pdf

An anelastic allspeed projection method for gravitationally stratified flows

Numerical model for the dynamics of a coupled fly line/fly rod system and experimental validation

This paper contributes a method to model the interactions of lowtension cables with the seabed. The cable is modeled as an elastica, and can support tension, torsion and two-axis bending. It is subjected to hydrodynamic forces as well as self-weight, buoyancy, and seabed contact. The seabed is modeled as an elastic foundation with linear damping and prescribed topology. A numerical algorithm is briefly described and then used to simulate cable laying. Several examples are studied including a cable towed in deep water, dropped on an uneven seabed, and finally, towed across an uneven seabed.

Numerical Simulations of Cable/seabed Interaction

This paper presents a conservative front-tracking method for shocks and contact discontinuities that is second-order accurate. It is based on a volume-of-fluid method that treats the moving front with concepts similar to those of an embedded-boundary method. Special care is taken in the centering of the data to ensure the right order of accuracy at the front, and a redistribution step guarantees conservation. A suite of test problems, for tracking both shocks and contact discontinuities, is presented that confirms that the method is second-order accurate.

/pdf/a-second-order-accurate-conservative-front-tracking-method-27ju9jepte.pdf

A Second-Order Accurate Conservative Front-Tracking Method in One Dimension

This paper presents an approximate analytical model of the dynamics of a tapered fly line during a standard overhead cast. During casting, the line forms a nonlinear propagating wave that is frequently referred to as a ‘loop’ by fly casters. The geometry of the loop is described by three distinct parts: a straight bottom segment (attached to a stationary fly rod), a semi-circular segment that is propagating (i.e. the loop), and a straight top segment that is also propagating, (i.e. the traveling line). A fly (particle) is attached at the end of the traveling line. A work-energy balance yields the velocity of the fly as a function of the length of the traveling line. For a uniform fly line (level line), a closed-form solution is found, while for a tapered fly line, the solution is obtained by quadrature. A critical loop diameter arises in the analysis, and it determines whether the final velocity of the fly is greater or lesser than the initial velocity of the traveling line. The analytical solutions are critically compared against numerical solutions of a general model for fly line dynamics that relaxes many of the assumptions employed in the analytical model. The agreement between the two solutions remains close during the loop propagation phase, provided a significant amount of fly line remains in the traveling line. However, as the traveling line vanishes and the loop ‘turns over’, the two solutions diverge abruptly due to the many simplifying assumptions employed in the analytical model.

Caroline Gatti-Bono

Papers

An anelastic allspeed projection method for gravitationally stratified flows

Numerical model for the dynamics of a coupled fly line/fly rod system and experimental validation

Numerical Simulations of Cable/seabed Interaction

A Second-Order Accurate Conservative Front-Tracking Method in One Dimension

Comparison of numerical and analytical solutions for fly casting dynamics