Accuracy of Shock Capturing in Two Spatial Dimensions

Carpenter, Mark H.; Casper, Jay

doi:10.2514/2.835

Cited by 96 publications

(40 citation statements)

References 13 publications

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“…This result is consistent with the finding of Ref. 32 where the spatial accuracy of the shock-capturing schemes reverted to first order behind the shock on suiliciently refined meshes, In ttds case, the first and second order error coefficients are of opposite sign, predicting error cancellation at the cross-over point (h= 7). The non-monotone behavior predicted from the emor analysis (using the three finest mesh solutions only) is qualitatively seen in the dkcrete solutions on the coarse meshes.…”

Section: )supporting

confidence: 81%

Verification and validation for laminar hypersonic flowfields

Roy

2000

Fluids 2000 Conference and Exhibit

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4bst!3@The accuracy .of Mach 8 kiminar flow solutions over a spherical y-blunted cone is verifiect, and the various physical models are validated. Verification of the solution accuracy is demonstrated by monitoring iterative convergence, performing a comprehensive grid convergence study, comparison to benchmark inviscid results, and code-to-code comparisons. Although the numerical scheme is nominally second order accurate in space, the presence of first order accuracy at the shock discontinuity results in first order behavior for the surface pressure distributions as the grid is suftlciently refined. Akernative methods are proposed for analyzing the spatial convergence behavior and determining the order of accuracy for mixed first and second order schemes. Validation of the wmputatiomd model is performed via surface pressure comparisons with high-quality experimental data. Careful attention is paid to the assumptions in the computational and experimental modeling. In particular, the thermodynamic state of the hypersonic wind tunnel nozzle is examined and arguments are made for the presence of a significant amount of thermal nonequilibrimn. Bias errors in the experimental Mach number calibration are discussed which are related to the assumption of thermal equilibrium (-0.23%) and the averaging of probe data over the entire test section (+0.6%). Differences between the simula- tion results for surface pressure and the experimental data are found to be as large as 3.3%. These differences are well outside the experimental 20 error bounds, even after accounting for the experimental bias errors.

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Section: )supporting

confidence: 81%

Verification and validation for laminar hypersonic flowfields

Roy

2000

Fluids 2000 Conference and Exhibit

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“…However, recently it has been reported (see [1], [2], [5]) that solutions of conservation laws obtained by formally high order accurate schemes degenerate to first order downstream of a shock layer. The effect is seen only when (i) the characteristics come out of the shock region and (ii) the solution is nonconstant.…”

Section: Keymentioning

confidence: 99%

Elimination of First Order Errors in Shock Calculations

Kreiss¹,

Efraimsson²,

Nordström³

2001

SIAM J. Numer. Anal.

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Abstract. First order errors downstream of shocks have been detected in computations with higher order shock capturing schemes in one and two dimensions. Based on a matched asymptotic expansion analysis we show how to modify the artificial viscosity and raise the order of accuracy. [20]). The small scale behavior is of significance in the applications above, and the phenomena involved cannot be captured without an appropriate resolution of this fine scale. The efficiency [10] of high order finite difference methods compared to low order methods can, of course, also be used to reduce the computational cost for a given level of accuracy. KeyHowever, recently it has been reported (see [1], [2], [5]) that solutions of conservation laws obtained by formally high order accurate schemes degenerate to first order downstream of a shock layer. The effect is seen only when (i) the characteristics come out of the shock region and (ii) the solution is nonconstant. Examples in one space dimension where both these conditions are satisfied include steady state calculations for systems with a source term and time dependent calculations for systems with nonconstant initial data.The downstream degeneration of accuracy has also been observed in calculations of acoustic waves in the presence of shocks. This case can be modeled by a scalar, linear equation where the wave speed changes value in a thin region without changing sign (see [2] and references therein). The degeneracy in accuracy is troublesome, even though the first order terms for reasonable mesh-sizes seem to be small in many cases.In [4], an explanation of the degeneracy in accuracy on solutions of conservation laws were given. Steady state solutions to systems with source terms were analyzed by using matched asymptotic expansions for a corresponding viscous equation, the so-called modified equation; see [13]. It was assumed that an inner and an outer solution exist. The inner solution is valid in the shock region and the outer solution elsewhere. The two solutions are matched in a so-called matching zone. The inner solution, the outer solution, and the shock position are expanded in powers of the small viscosity coefficient ε. With ε = O(h) in the shock region, the outer solution contains a term of O(h) downstream of the shock. Upstream this term is zero.

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“…Some of the reasons are as follows: (1) a programming error exists in the computer code; (2) the numerical algorithm is deficient is some unanticipated way; (3) there is insufficient grid resolution such that the grid is not in the asymptotic convergence region of the power-series expansion for the particular system response quantity (SRQ) of interest, (4) the formal order of convergence for interior grid points is different from the formal order of convergence for boundary conditions involving derivatives, resulting in a mixed order of convergence over the solution domain; (5) singularities, discontinuities, and contact surfaces are interior to the domain of the PDE; (6) singularities and discontinuities occur along the boundary of the domain; (7) the mesh resolution changes abruptly over the solution domain; (8) there is inadequate convergence of an iterative procedure in the numerical algorithm; and (9) boundary conditions are overspecified. It is beyond the scope of this paper to discuss the reasons listed above in detail; however, some of the representative references in these topics are [1,33,44,[47][48][49][50][51][52][53][54][55][56].…”

Section: Figure 3 Observed Order Of Convergence As a Function Of Meshmentioning

confidence: 99%

Verification and validation benchmarks

Oberkampf

Trucano²

2008

Nuclear Engineering and Design

181

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Verification and validation (V&V) are the primary means to assess the accuracy and reliability of computational simulations. V&V methods and procedures have fundamentally improved the credibility of simulations in several high-consequence fields, such as nuclear reactor safety, underground nuclear waste storage, and nuclear weapon safety. Although the terminology is not uniform across engineering disciplines, code verification deals with assessing the reliability of the software coding, and solution verification deals with assessing the numerical accuracy of the solution to a computational model. Validation addresses the physics modeling accuracy of a computational simulation by comparing the computational results with experimental data. Code verification benchmarks and validation benchmarks have been constructed for a number of years in every field of computational simulation. However, no comprehensive guidelines have been proposed for the construction and use of V&V benchmarks. For example, the field of nuclear reactor safety has not focused on code verification benchmarks, but it has placed great emphasis on developing validation benchmarks. Many of these validation benchmarks are closely related to the operations of actual reactors at near-safety-critical conditions, as opposed to being more fundamental-physics benchmarks. This paper presents recommendations for the effective design and use of code verification benchmarks based on manufactured solutions, classical analytical solutions, and highly accurate numerical solutions. In addition, this paper presents recommendations for the design and use of validation benchmarks, highlighting the careful design of building-block experiments, the estimation of experimental measurement uncertainty for both inputs and outputs to the code, validation metrics, and the role of model calibration in validation. It is argued that the understanding of predictive capability of a computational model is built on the level of achievement in V&V activities, how closely related the V&V benchmarks are to the actual application of interest, and the quantification of uncertainties related to the application of interest.

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Accuracy of Shock Capturing in Two Spatial Dimensions

Cited by 96 publications

References 13 publications

Verification and validation for laminar hypersonic flowfields

Verification and validation for laminar hypersonic flowfields

Elimination of First Order Errors in Shock Calculations

Verification and validation benchmarks

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