Rupert Klein

A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.

Semi-implicit extension of a godunov-type scheme based on low mach number asymptotics I: One-dimensional flow

A single scale, multiple space scale asymptotic analysis provides detailed insight into the low Mach number limit behavior of solutions of the compressible Euler equations. We use the asymptotics as a guideline for developing a low Mach number extension of an explicit higher order shock-capturing scheme. This semi-implicit scheme involves multiple pressure variables, large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis. Advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step. This equation is a pressure correction equation for incompressible flows when the Mach number is zero. In the low Mach number limit, the time step is restricted by a Courant number based essentially on the maximum flow velocity. For moderate and large Mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm.

Systematic Multiscale Models for the Tropics

Systematic multi-scale perturbation theory is utilized to develop self-consistent simplified model equations for the interaction across multiple spatial and/or temporal scales in the tropics. One of these models involves simplified equations for intraseasonal planetary equatorial synoptic scale dynamics (IPESD). This model includes the self-consistent quasi-linear interaction of synoptic scale generalized steady Matsuno-Webster-Gill models with planetary scale dynamics of equatorial long waves. These new models have the potential for providing self-consistent prognostic and diagnostic models for the intraseasonal tropical oscillation. Other applications of the systematic approach reveal three different balanced weak temperature gradient (WTG) approximations for the tropics with different regimes of validity in space and time: a synoptic equatorial scale WTG (SEWTG), a mesoscale equatorial WTG (MEWTG) which reduces to the classical models treated by others, and a new seasonal planetary equatorial WTG (SPEWTG). Both the SPEWTG and MEWTG model equations have solutions with general vertical structure yet have the linearized dispersion relation of barotropic Rossby waves; thus, these models can play an important role in theories for midlatitude connections with the tropics. The models are derived both from the equatorial shallow water equations in a simplified context and also as distinguished limits from the compressible primitive equations in general.

Asymptotic adaptive methods for multi-scale problems in fluid mechanics

This paper reports on the results of a three-year research effort aimed at investigating and exploiting the role of physically motivated asymptotic analysis in the design of numerical methods for singular limit problems in fluid mechanics. Such problems naturally arise, among others, in combustion, magneto-hydrodynamics, and geophysical fluid mechanics. Typically, they are characterized by multiple-space and/or -time scales and by the disturbing fact that standard computational techniques fail entirely, are unacceptably expensive, or both. The challenge here is to construct numerical methods which are robust, uniformly accurate, and efficient through different asymptotic regimes and over a wide range of relevant applications. Summaries of multiple-scales asymptotic analyses for low-Mach-number flows, magneto-hydrodynamics at small Mach and Alfven numbers, and of multiple-scales atmospheric flows are provided. These reveal singular balances between selected terms in the respective governing equations within the considered flow regimes. These singularities give rise to problems of severe stiffness, stability, or to dynamic-range issues in straight-forward numerical discretizations. A formal mathematical framework for the multiple scales asymptotics is then summarized by use of the example of multiple-length-scale single-time-scale asymptotics for low-Mach-number flows. The remainder of the paper focuses on the construction of numerical discretizations for the respective full governing equation systems. These discretizations avoid the pitfalls of singular balances by exploiting the asymptotic results. Importantly, the asymptotics are not used here to derive simplified equation systems, which are then solved numerically. Rather, numerical integration of the full equation sets is aimed at and the asymptotics are used only to construct discretizations that do not deteriorate as a singular limit is approached. One important ingredient of this strategy is the numerical identification of a singular limit regime given a set of discrete numerical state variables. This problem is addressed in an exemplary fashion for multiple-length single-time-scale low-Mach-number flows in one space dimension. The strategy allows a dynamic determination of an instantaneous relevant Mach number, and it can thus be used to drive the appropriate adjustment of the numerical discretizations when the singular limit regime is approached.

A capturing - tracking hybrid scheme for deflagration discontinuities

A new numerical technique for the simulation of gas dynamic discontinuities in compressible flows is presented. The schemes complexity and structure is intermediate between a higher-order shock-capturing technique and a front-tracking algorithm. It resembles a tracking scheme in that the front geometry is explicitly computed using a level set method. However, we employ the geometrical information gained in an unusual fashion. Instead of letting it define irregular part-cells wherever the front intersects a grid cell of the underlying mesh and separately balancing fluxes for these part-cells, we use the information for an accurate reconstruction of the discontinous solution in these mixed cells. The reconstructed states and again the front geometry are then used to define accurate effective numerical fluxes across those regular grid cell interfaces that are intersected by the front during the time step considered. Hence, the scheme resembles a capturing scheme in that only cell averages of conserved quantities for full cells of the underlying grid are computed. A side-effect is that the small subcell CFL problem of other conservative tracking schemes is eliminated. A disadvantage for certain applications is that the scheme is conservative with respect to the underlying grid, but that it is not separately conservative with respect to the pre- and post-front regions. If this is a crucial requirement, additional measures have to be taken.

The extension of incompressible flow solvers to the weakly compressible regime

Numerical simulation schemes for incompressible flows such as the SIMPLE scheme are extended to weakly compressible fluid flow. A single time scale, multiple space scale asymptotic analysis is used to gain insight into the limit behavior of the compressible flow equations as the Mach number vanishes. Motivated by these results, multiple pressure variables (MPV) are introduced into the numerical framework. These account separately for thermodynamic effects, acoustic wave propagation and the balance of forces. Discretized analogues of the averaging and large scale differencing procedures known from multiple scales asymptotics allow accurate capturing of various physical phenomena that are operative on very different length scales. The MPV approach combines the explicit numerical computation of global compression from the boundary and the long wavelength acoustics on coarse grids with an implicit pressure or pressure correction equation that formally converges to the corresponding incompressible one when the Mach number tends to zero

Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods

Prandtl’s boundary layer theory may be considered one of the origins of systematic scale analysis and asymptotics in fluid mechanics. Due to the vast scale differences in atmospheric flows such analyses have a particularly strong tradition in theoretical meteorology. Simplified asymptotic limit equations,derive d through scale analysis,yield a deep insight into the dynamics of the atmosphere. Due to limited capacities of even the fastest computers, the use of such simplified equations has traditionally been a necessary precondition for successful approaches to numerical weather forecasting and climate modelling. In the face of the continuing increase of available compute power there is now a strong tendency to relax as many simplifying scaling assumptions as possible and to go back to more complete and more complex balance equations in atmosphere flow computations. However,the simplified equations obtained through scaling analyses are generally associated with singular asymptotic limits of the full governing equations,and this has important consequences for the numerical integration of the latter. In these singular limit regimes dominant balances of a few terms in the governing equations lead to degeneracies and singular changes of the mathematical structure of the equations. Numerical models based on comprehensive equation systems must simultaneously represent these dominant balances and the subtle,but important,deviations from them. These requirements are partly in contradiction,and this can lead to severe restrictions of the accuracy and/or efficiency of numerical models. The present paper makes a case for a somewhat unconventional use of the results of scale analyses and multiple scales asymptotics. It demonstrates how,thr ough the judicious implementation of asymptotic results,numeric al discretizations of the full governing equations can be designed so that they operate with uniform accuracy and efficiency even when a singular limit regime is approached.

A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces

We present a finite volume method for the solution of the two-dimensional elliptic equation ??(s(x)?u(x))=f(x) with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution u(x) resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the L∞ and L2 norm.

Simplified equations for the interaction of nearly parallel vortex filaments

New simplified asymptotic equations for the interaction of nearly parallel vortex filaments are derived and analysed here. The simplified equations retain the important physical effects of linearized local self-induction and nonlinear potential vortex interaction among different vortices but neglect other non-local effects of self-stretching and mutual induction. These equations are derived systematically in a suitable distinguished asymptotic limit from the Navier–Stokes equations. The general Hamiltonian formalism and conserved quantities for the simplified equations are developed here. Properties of these asymptotic equations for the important special case involving nearly parallel pairs of interacting filaments are developed in detail. In particular, strong evidence is presented that for any filament pair with a negative circulation ratio, there is finite-time collapse in a self-similar fashion independent of the perturbation but with a structure depending on the circulation ratio. On the other hand, strong evidence is presented that no finite-time collapse is possible for perturbations of a filament pair with a positive circulation ratio. The present theory is also compared and contrasted with earlier linear and nonlinear stability analyses for pairs of filaments.

Self-stretching of a perturbed vortex filament I: the asymptotic equation for deviations from a straight line

A new asymptotic equation is derived for the motion of thin vortex filaments in an incompressible fluid at high Reynolds numbers. This equation differs significantly from the familiar local self-induction equation in that it includes self-stretching of the filament in a nontrivial, but to some extent analytically tractable, fashion. Under the same change of variables as employed by Hasimoto (1972) to convert the local self-induction equation to the cubic nonlinear Schrodinger equation, the new asymptotic propagation law becomes a cubic nonlinear Schrodinger equation perturbed by an explicit nonlocal, linear operator. Explicit formulae are developed which relate the rate of local self-stretch along the vortex filament to a particular quadratic functional of the solution of the perturbed Schrodinger equation. The asymptotic equation is derived systematically from suitable solutions of the Navier-Stokes equations by the method of matched asymptotic expansions based on the limit of high Reynolds numbers. The key idea in the derivation is to consider a filament whose core deviates initially from a given smooth curve only by small-amplitude but short-wavelength displacements balanced so that the axial length scale of these perturbations is small compared to an integral length of the background curve but much larger than a typical core size δ=O(Re-1/2) of the filament. In a particular distinguished limit of wavelength, preturbation amplitude and filament core size the nonlocal induction integral has a simplified asymptotic representation and yields a contribution in the Schrodinger equation that directly competes with the cubic nonlinearity.