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

Banks, Jeffrey W.; Aslam, Tariq D.; Rider, William J.

doi:10.1016/j.jcp.2008.04.002

Cited by 95 publications

(82 citation statements)

References 29 publications

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“…However, the velocity v converges as O(h 2 x ) in the L 1 -norm but the degraded rate of O(h 4/3 x ) for the L ∞ -norm. This somewhat surprising behavior is the result of the sharp switch introduced by the MinMod limiter and is similar to behavior exhibited by high-resolution schemes for the first-order system [19].…”

Section: One-dimensional Traveling Sine Wavesupporting

confidence: 69%

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Upwind schemes for the wave equation in second-order form

Banks

Henshaw

2012

Journal of Computational Physics

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We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d'Alembert's exact solution. We construct conservative finite difference approximations, although finite volume approximations are also possible. High-order accuracy is obtained using a space-time procedure which requires only two discrete time levels. The advantages of our approach include efficiency in both memory and speed together with accuracy and robustness. The stability and accuracy of the approximations in one and two space dimensions are studied through normal-mode analysis. The form of the dissipation and dispersion introduced by the schemes is elucidated from the modified equations. Upwind schemes are implemented and verified in one dimension for approximations up to sixth-order accuracy, and in two dimensions for approximations up to fourth-order accuracy. Numerical computations demonstrate the attractive properties of the approach for solutions with varying degrees of smoothness.

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Section: One-dimensional Traveling Sine Wavesupporting

confidence: 69%

“…All the schemes under consideration converge at the rate O(h p/p+1 x ) where p is the nominal convergence rate for the scheme for smooth problems. These are the expected convergence rates for numerical approximations to the first-order system with jump initial data [19].…”

Section: One-dimensional Top-hat Problemmentioning

confidence: 92%

Upwind schemes for the wave equation in second-order form

Banks

Henshaw

2012

Journal of Computational Physics

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“…In Figure 4, plots of the approximate solution and error show that the error field has singularities where the solution crosses through zero, i.e., where the upwind scheme changes its stencil. Degraded convergence is well known in the presence of singularities [28]. We note that the error estimate is still asymptotically correct in any L p -norm.…”

Section: Properties For Non-smooth Discretizationsmentioning

confidence: 64%

“…The large error introduced near the extrema impacts the L ∞ error norm most, and we see convergence rates that trend towards a value of 4/3 . Interestingly, this is the expected rate of convergence near a slope discontinuity for a second order scheme [28]. The L 1 error norm is less sensitive and asymptotically tends to the rate of two.…”

Section: Error Evolution For Nonlinear Discretizationsmentioning

confidence: 73%

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Banks

Hittinger

Connors

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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The estimation of discretization error in numerical simulations is a key component in the development of uncertainty quantification. In particular, there exists a need for reliable, robust estimators for finite volume and finite difference discretizations of hyperbolic partial differential equations. The approach espoused here, often called the error transport approach in the literature, is to solve an auxiliary error equation concurrently with the primal governing equation to obtain a point-wise (cell-wise) estimate of the discretization error. One-dimensional, nonlinear, time-dependent problems are considered. In contrast to previous work, fully nonlinear error equations are advanced, and potential benefits are identified. A systematic approach to approximate the local residual for both method-of-lines and space-time discretizations is developed. Behavior of the error estimates on problems that include weak solutions demonstrates that this nonlinear error evolution provides useful error estimates.

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“…This was likely due to the complexity of the flow physics. It has been noted in literature that in the presence of discontinuities, non-monotonic behaviour such as that shown in the Figure 3.4 is to be expected (Oberkampf and Roy, 2010) and no matter what the solver's formal solution order, the solution order becomes linear in the vicinity of shock waves (Banks et al, 2008).…”

Section: Grid Convergence Behaviourmentioning

confidence: 84%

Mixing and combustion enhancement in a Mach 12 shape-transitioning scramjet engine

Barth¹

118

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are to investigate and characterize the flow physics behind the Mach 12 REST engine's current performance, and then attempt to improve its combustion performance by tailoring the engine's fuel injection to its internal flow field without otherwise modifying the engine geometry. To meet these aims, the engine was studied both numerically and experimentally.The first-ever combusting simulations of a REST scramjet operating at Mach 12 conditions were performed for the Mach 12 REST engine using the CFD research code US3D. The simulations covered a range of conditions, including: unfuelled engine flow, inlet-fuelled flow, and various combined inlet/combustor fuelling configurations. The simulations were found to match well with the experiments they were designed to reproduce and be compared against. A comparison of simulations with experimental inflow conditions and their equivalent flight conditions on an otherwise identical engine showed that experiments in the T4 Stalker tube reproduce engine pressure and heat flux distributions well. The tunnel condition tends to capture less incoming flow than the engine at flight conditions, which leads to the ground-tested engine over-predicting the engine equivalence ratio.The Mach 12 REST inlet was found to produce a thick "bubble-shaped" boundary layer along its bodyside compression surface, due to the compression effects of the inlet sidewalls acting on a thick turbulent boundary layer ingested from the vehicle forebody. This thick boundary layer forces the majority of inlet-captured air into a high-density, high Mach number flow region along the engine cowlside wall . The inlet also produces a symmetric pair of high-temperature swept separations that enter the engine isolator along the sidewalls of the engine. When fuel is injected from the bodyside surface of the inlet, it remains trapped inside the thick, turbulent boundary layer, where it becomes well-mixed and begins to burn just upstream of the inlet throat.i As much as 50% of this fuel is burned by the time it enters the engine isolator, while its injection and burning increases the inlet's drag by less than 5%. This burning bodyside flow region thermally compresses the remaining air flow within the engine isolator, and provides a source of heat and combustion radicals for the ignition of fuel injected further downstream. Overall combustion efficiency of inlet-injected fuel at the engine exhaust plane was found to be nearly 80% at high equivalence ratios.Flow within the Mach 12 REST combustor is strongly shock-dominated. This is caused by both the cowl closure shock train transmitted from the inlet, and a strong recompression shock generated at the start of the combustor. This recompression shock is generated by the flow passing over a backward step at the entrance to the combustor, and is reinforced on the cowlside of the engine by compression caused by the engine flow path turning to realign with the nominal direction of flight. Fuel injected from the face of the combustor step was combined with inlet injection in...

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

Cited by 95 publications

References 29 publications

Upwind schemes for the wave equation in second-order form

Upwind schemes for the wave equation in second-order form

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Mixing and combustion enhancement in a Mach 12 shape-transitioning scramjet engine

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