Jeffrey W. Banks

On sub-linear convergence for linearly degenerate waves in capturing schemes

A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t^1^/^2, yielding a convergence rate of 1/2. Self-similar heuristics for higher-order discretizations produce a growth rate for the layer thickness of @Dt^1^/^(^p^+^1^) which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

A numerical method is described for inviscid, compressible, multi-material flow in two space dimensions. The flow is governed by the multi-material Euler equations with a general mixture equation of state. Composite overlapping grids are used to handle complex flow geometry and block-structured adaptive mesh refinement (AMR) is used to locally increase grid resolution near shocks and material interfaces. The discretization of the governing equations is based on a high-resolution Godunov method, but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard, conservative shock-capturing schemes. The energy correction is constructed based on a uniform-pressure-velocity flow and is significant only near the captured interface. A variety of two-material flows are presented to verify the accuracy of the numerical approach and to illustrate its use. These flows assume an equation of state for the mixture based on the Jones-Wilkins-Lee (JWL) forms for the components. This equation of state includes a mixture of ideal gases as a special case. Flow problems considered include unsteady one-dimensional shock-interface collision, steady interaction of a planar interface and an oblique shock, planar shock interaction with a collection of gas-filled cylindrical inhomogeneities, and the impulsive motion of the two-component mixture in a rigid cylindrical vessel.

Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods

This paper explores the development of a scalable, nonlinear, fully-implicit stabilized unstructured finite element (FE) capability for 2D incompressible (reduced) resistive MHD. The discussion considers the implementation of a stabilized FE formulation in context of a fully-implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton-Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, including both low-Lundquist number (e.g., an MHD Faraday conduction pump and a hydromagnetic Rayleigh-Bernard linear stability calculation) and moderately-high Lundquist number (magnetic island coalescence problem) examples. Initial results that explore the scaling of the solution methods are presented on up to 4096 processors for problems with up to 64M unknowns on a CrayXT3/4. Additionally, a large-scale proof-of-capability calculation for 1 billion unknowns for the MHD Faraday pump problem on 24,000 cores is presented.

A New Class of Nonlinear Finite-Volume Methods for Vlasov Simulation

Methods for the numerical discretization of the Vlasov equation should efficiently use the phase-space discretization and should introduce only enough numerical dissipation to promote stability and control oscillations. A new high-order nonlinear finite-volume algorithm for the Vlasov equation that discretely conserves particle number and controls oscillations is presented. The method is fourth order in space and time in well-resolved regions but smoothly reduces to a third-order upwind scheme as features become poorly resolved. The new scheme is applied to several standard problems for the Vlasov-Poisson system, and the results are compared with those from other finite-volume approaches, including an artificial viscosity scheme and the piecewise parabolic method. It is shown that the new scheme is able to control oscillations while preserving a higher degree of fidelity of the solution than the other approaches.

An analysis of a new stable partitioned algorithm for FSI problems. Part I: Incompressible flow and elastic solids

Stable partitioned algorithms for fluid–structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. Importantly, these new added-mass partitioned (AMP) schemes are stable and retain full accuracy with no sub-iterations per time step, even in the presence of strong added-mass effects (e.g. for light solids). The numerical approach described here for bulk compressible solids extends the scheme of Banks et al. [1,2] for inviscid compressible flow, and uses Robin (mixed) boundary conditions with the fluid and solid solvers at the interface. The basic AMP Robin conditions, involving a linear combination of velocity and stress, are determined from the outgoing solid characteristic relation normal to the fluid–solid interface combined with the matching conditions on the velocity and traction. Two alternative forms of the AMP conditions are then derived depending on whether the fluid equations are advanced with a fractional-step method or not. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems; the full nonlinear case being left for future consideration. A normal mode analysis is performed to show that the new AMP algorithm is stable for any ratio of the solid and fluid densities, including the case of very light solids when added-mass effects are large. In contrast, it is shown that a traditional partitioned algorithm involving a Dirichlet–Neumann coupling for the same FSI problem is formally unconditionally unstable for any ratio of densities. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy solids.

Deforming composite grids for solving fluid structure problems

We describe a mixed Eulerian-Lagrangian approach for solving fluid-structure interaction (FSI) problems. The technique, which uses deforming composite grids (DCG), is applied to FSI problems that couple high speed compressible flow with elastic solids. The fluid and solid domains are discretized with composite overlapping grids. Curvilinear grids are aligned with each interface and these grids deform as the interface evolves. The majority of grid points in the fluid domain generally belong to background Cartesian grids which do not move during a simulation. The FSI-DCG approach allows large displacements of the interfaces while retaining high quality grids. Efficiency is obtained through the use of structured grids and Cartesian grids. The governing equations in the fluid and solid domains are evolved in a partitioned approach. We solve the compressible Euler equations in the fluid domains using a high-order Godunov finite-volume scheme. We solve the linear elastodynamic equations in the solid domains using a second-order upwind scheme. We develop interface approximations based on the solution of a fluid-solid Riemann problem that results in a stable scheme even for the difficult case of light solids coupled to heavy fluids. The FSI-DCG approach is verified for three problems with known solutions, an elastic-piston problem, the superseismic shock problem and a deforming diffuser. In addition, a self convergence study is performed for an elastic shock hitting a fluid filled cavity. The overall FSI-DCG scheme is shown to be second-order accurate in the max-norm for smooth solutions, and robust and stable for problems with discontinuous solutions for a wide range of constitutive parameters.

An analysis of a new stable partitioned algorithm for FSI problems. Part II: Incompressible flow and structural shells

Stable partitioned algorithms for fluid–structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. The numerical approach described here for structural shells uses Robin (mixed) interface conditions for the pressure and velocity in the fluid which are derived directly from the governing equations. The resulting added-mass partitioned (AMP) algorithm is stable even for very light structures, requires no sub-iterations per time step, and is second-order accurate. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems. A normal mode analysis is performed to show that the new AMP algorithm is stable, even for the case of very light structures when added-mass effects are large. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy structures. A summary comparison of the AMP algorithm developed here and the one in Part I is provided.

A Study of Detonation Propagation and Diffraction with Compliant Confinement

Previous computational studies of diffracting detonations with the ignition-and-growth (IG) model demonstrated that, contrary to experimental observations, the computed solution did not exhibit dead zones. For a rigidly confined explosive it was found that while diffraction past a sharp corner did lead to a temporary separation of the lead shock from the reaction zone, the detonation re-established itself in due course and no pockets of unreacted material remained. The present investigation continues to focus on the potential for detonation failure within the IG model, but now for a compliant confinement of the explosive. The aim of the present paper is two-fold. First, in order to compute solutions of the governing equations for multi-material reactive flow, a numerical method is developed and discussed. The method is a Godunov-type, fractional-step scheme which incorporates an energy correction to suppress numerical oscillations that occur near material interfaces for standard conservative schemes. The accuracy of the solution method is then tested using a two-dimensional rate-stick problem for both strong and weak confinements. The second aim of the paper is to extend the previous computational study of the IG model by considering two related problems. In the first problem, the corner-turning configuration is re-examined, and it is shown that in the matter of detonation failure, the absence of rigid confinement does not affect the outcome in a material way; sustained dead zones continue to elude the model. In the second problem, detonations propagating down a compliantly confined pencil-shaped configuration are computed for a variety of cone angles of the tapered section. It is found, in accord with experimental observation, that if the cone angle is small enough, the detonation fails prior to reaching the cone tip. For both the corner-turning and the pencil-shaped configurations, mechanisms underlying the behaviour of the computed solutions are identified.

Two-dimensional Vlasov simulation of electron plasma wave trapping, wavefront bowing, self-focusing, and sideloss

Two-dimensional Vlasov simulations of nonlinear electron plasma waves are presented, in which the interplay of linear and nonlinear kinetic effects is evident. The plasma wave is created with an external traveling wave potential with a transverse envelope of width Δy such that thermal electrons transit the wave in a “sideloss” time, tsl~Δy/ve. Here, ve is the electron thermal velocity. The quasisteady distribution of trapped electrons and its self-consistent plasma wave are studied after the external field is turned off. In cases of particular interest, the bounce frequency, ωbe=keϕ/me, satisfies the trapping condition ωbetsl>2π such that the wave frequency is nonlinearly downshifted by an amount proportional to the number of trapped electrons. Here, k is the wavenumber of the plasma wave and ϕ is its electric potential. For sufficiently short times, the magnitude of the negative frequency shift is a local function of ϕ. Because the trapping frequency shift is negative, the phase of the wave on axis lags ...

A stable FSI algorithm for light rigid bodies in compressible flow

Abstract In this article we describe a stable partitioned algorithm that overcomes the added mass instability arising in fluid–structure interactions of light rigid bodies and inviscid compressible flow. The new algorithm is stable even for bodies with zero mass and zero moments of inertia. The approach is based on a local characteristic projection of the force on the rigid body and is a natural extension of the recently developed algorithm for coupling compressible flow and deformable bodies [1] , [2] , [3] . The new algorithm advances the solution in the fluid domain with a standard upwind scheme and explicit time-stepping. The Newton–Euler system of ordinary differential equations governing the motion of the rigid body is augmented by added mass correction terms. This system, which is very stiff for light bodies, is solved with an A-stable diagonally implicit Runge–Kutta scheme. The implicit system (there is one independent system for each body) consists of only 3 d + d 2 scalar unknowns in d = 2 or d = 3 space dimensions and is fast to solve. The overall cost of the scheme is thus dominated by the cost of the explicit fluid solver. Normal mode analysis is used to prove the stability of the approximation for a one-dimensional model problem and numerical computations confirm these results. In multiple space dimensions the approach naturally reveals the form of the added mass tensors in the equations governing the motion of the rigid body. These tensors, which depend on certain surface integrals of the fluid impedance, couple the translational and angular velocities of the body. Numerical results in two space dimensions, based on the use of moving overlapping grids and adaptive mesh refinement, demonstrate the behavior and efficacy of the new scheme. These results include the simulation of the difficult problems of shock impingement on an ellipse and a more complex body with appendages, both with zero mass.