scholarly journals A SEMI-ANALYTIC EIGENVALUE EXTENSION TO THE DOPPLER SLAB ANALYTIC BENCHMARK1

2021 ◽  
Vol 247 ◽  
pp. 04018
Author(s):  
Kyle E. Remley ◽  
David P. Griesheimer
Keyword(s):  
Numerical Computation ◽  
Eigenvalue Problems ◽  
Analytic Solutions ◽  
Numerical Implementation ◽  
Numerical Schemes ◽  
Multiphysics Simulation ◽  
Reactivity Coefficient ◽  
Coupled Solvers ◽  
Benchmark Solutions

Advancement in multiphysics simulation has motivated interest in availability of analytic and semi-analytic benchmark solutions. These solutions are sought because they can be used to assess the accuracy of complicated numerical schemes necessary to simulate coupled physics systems. While there exist analytic solutions for fixed-source problems, benchmark-quality eigenvalue solutions are of interest because eigenvalue problems more closely align with analyses undertaken with coupled solvers. This paper extends a fixed-source benchmark, the Doppler Slab benchmark, to the eigenvalue case. A novel solution for this benchmark is derived. Numerical implementation of the benchmark is demonstrated through verification of numerical computation of the power reactivity coefficient.

VALIDATING RANGE-DEPENDENT, FULL-FIELD MODELS OF THE ACOUSTIC VECTOR FIELD IN SHALLOW WATER ENVIRONMENTS

2008 ◽  
Vol 16 (04) ◽  
pp. 471-486 ◽  
Author(s):  
KEVIN B. SMITH
Keyword(s):  
Parabolic Equation ◽  
Normal Mode ◽  
Numerical Algorithms ◽  
Analytic Solutions ◽  
Equation Model ◽  
Propagation Model ◽  
Full Field ◽  
Benchmark Solutions ◽  
Fourier Algorithm ◽  
Good Agreement

Numerical algorithms for computing acoustic particle velocity from a pressure propagation model are introduced. Implementation using both a parabolic equation and normal mode approach are considered. The parabolic equation model employed uses a split-step Fourier algorithm, although application of the technique is general to other parabolic equation models. Expressions for the normal mode equations are also presented, for both coupled and adiabatic mode models. Results for a Pekeris waveguide are presented for a point source, prompting a brief discussion of multipath influence on the estimation of the direction of energy flow. Approximate analytic solutions are used to validate the general results of both the models. Results for the range-dependent benchmark wedge are then presented, which show generally good agreement between the two types of models. The results from the two-way, coupled normal mode model provide potential benchmark solutions for the wedge and a means of confirming the accuracy of other models.

scholarly journals On the numerical schemes for Langevin-type equations

2020 ◽  
Vol 99 (3) ◽  
pp. 62-74
Author(s):  
M. Akat ◽  
◽  
R. Kosker ◽  
A. Sirma ◽  
◽  
...  
Keyword(s):  
Numerical Scheme ◽  
Numerical Approach ◽  
Linear Case ◽  
Numerical Implementation ◽  
Numerical Schemes ◽  
Variation Of Constants Formula ◽  
Variation Of Constants ◽  
Non Linear ◽  
Numerical Discretization ◽  
The Stability

In this paper, a numerical approach is proposed based on the variation-of-constants formula for the numerical discretization Langevin-type equations. Linear and non-linear cases are treated separately. The proofs of convergence have been provided for the linear case, and the numerical implementation has been executed for the non-linear case. The order one convergence for the numerical scheme has been shown both theoretically and numerically. The stability of the numerical scheme has been shown numerically and depicted graphically.

scholarly journals SUPER-HISTORY METHODS FOR ADJOINT-WEIGHTED TALLIES IN MONTE CARLO TIME EIGENVALUE CALCULATIONS

2021 ◽  
Vol 247 ◽  
pp. 04008
Author(s):  
F. Filiciotto ◽  
A. Jinaphanh ◽  
A. Zoia
Keyword(s):  
Monte Carlo ◽  
Eigenvalue Problems ◽  
Neutron Transport ◽  
Computation Time ◽  
Nuclear Data ◽  
Analytic Solutions ◽  
Benchmark Problems ◽  
Probability Method ◽  
Fission Probability ◽  
Transport Problems

Time eigenvalues emerge in several key applications related to neutron transport problems, including reactor start-up and reactivity measurements. In this context, experimental validation and uncertainty quantification would demand to assess the variation of the dominant time eigenvalue in response to a variation of nuclear data. Recently, we proposed the use of a Generalized Iterated Fission Probability method (G-IFP) to compute adjoint-weighted tallies, such as kinetic parameters, perturbations and sensitivity coefficients, for Monte Carlo time (or alpha) eigenvalue calculations. With the massive use of parallel Monte Carlo calculations, it would be therefore useful to trade the memory burden of the G-IFP method (which is comparable to that of the standard IFP method for k-eigenvalue problems) for computation time and to rely on history-based schemes for such adjoint-weighted tallies. For this purpose, we investigate the use of the super-history method as applied to estimating adjoint-weighted tallies within the α-k power iteration, based on previous work on k-eigenvalue problems. Verification of the algorithms is performed on some simple preliminary tests where analytic solutions exist. In addition, the performances of the proposed method are assessed by comparing the super-history and the G-IFP methods for the same sets of benchmark problems.

scholarly journals New analytic bending, buckling, and free vibration solutions of rectangular nanoplates by the symplectic superposition method

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Xinran Zheng ◽  
Mingqi Huang ◽  
Dongqi An ◽  
Chao Zhou ◽  
Rui Li
Keyword(s):  
Hamiltonian System ◽  
Free Vibration ◽  
Eigenvalue Problems ◽  
Analytic Solutions ◽  
Symplectic Space ◽  
Superposition Method ◽  
Mathematical Solution ◽  
Simply Supported ◽  
Solution Methods ◽  
Symplectic Eigenvalue

AbstractNew analytic bending, buckling, and free vibration solutions of rectangular nanoplates with combinations of clamped and simply supported edges are obtained by an up-to-date symplectic superposition method. The problems are reformulated in the Hamiltonian system and symplectic space, where the mathematical solution framework involves the construction of symplectic eigenvalue problems and symplectic eigen expansion. The analytic symplectic solutions are derived for several elaborated fundamental subproblems, the superposition of which yields the final analytic solutions. Besides Lévy-type solutions, non-Lévy-type solutions are also obtained, which cannot be achieved by conventional analytic methods. Comprehensive numerical results can provide benchmarks for other solution methods.

NUMERICAL SCHEMES OBTAINED FROM LATTICE BOLTZMANN EQUATIONS FOR ADVECTION DIFFUSION EQUATIONS

2006 ◽  
Vol 17 (11) ◽  
pp. 1563-1577 ◽  
Author(s):  
SHINSUKE SUGA
Keyword(s):  
Lattice Boltzmann ◽  
Peclet Number ◽  
Eigenvalue Problems ◽  
Velocity Model ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Péclet Number ◽  
Velocity Models ◽  
Advection Diffusion ◽  
The Stability

Stability and accuracy of the numerical schemes obtained from the lattice Boltzmann equation (LBE) used for numerical solutions of two-dimensional advection-diffusion equations are presented. Three kinds of velocity models are used to determine the moving velocity of particles on a squre lattice. A system of explicit finite difference equations are derived from the LBE based on the Bhatnagar, Gross and Krook (BGK) model for individual velocity model. In order to approximate the advecting velocity field, a linear equilibrium distribution function is used for each of the moving directions. The stability regions of the schemes in the special case of the relaxation parameter ω in the LBE being set to ω=1 are found by analytically solving the eigenvalue problems of the amplification matrices corresponding to each scheme. As for the cases of general relaxation parameters, the eigenvalue problems are solved numerically. A benchmark problem is solved in order to investigate the relationship between the accuracy of the numerical schemes and the order of the Peclet number. The numerical experiments result in indicating that for the scheme based on a 9-velocity model we can find the parameters depending on the order of the given Peclet number, which generate accurate solutions in the stability region.

Benchmark Solutions for Slip Flow and H1 Heat Transfer in Rectangular and Equilateral Triangular Ducts

2012 ◽  
Vol 135 (2) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Heat Transfer ◽  
Exact Solutions ◽  
Temperature Jump ◽  
Slip Flow ◽  
Velocity Slip ◽  
Analytic Solutions ◽  
Nusselt Numbers ◽  
Benchmark Solutions

Accurate analytic solutions for the fully developed slip flow and H1 heat transfer in rectangular and equilateral triangular ducts are presented. Both velocity slip and temperature jump have significant influences on the Poiseuille and Nusselt numbers. These exact solutions serve as benchmark cases for other methods, whether analytic, approximate, or numerical.

scholarly journals Schwarz waveform relaxation algorithm for heat equations with distributed delay

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 659-667
Author(s):  
Shu-Lin Wu
Keyword(s):  
Numerical Computation ◽  
Distributed Delay ◽  
Computation Time ◽  
Analytic Solutions ◽  
Transmission Conditions ◽  
Waveform Relaxation ◽  
Heat Equations ◽  
Relaxation Algorithm ◽  
Schwarz Waveform Relaxation ◽  
The One

Heat equations with distributed delay are a class of mathematic models that has wide applications in many fields. Numerical computation plays an important role in the investigation of these equations, because the analytic solutions of partial differential equations with time delay are usually unavailable. On the other hand, duo to the delay property, numerical computation of these equations is time-consuming. To reduce the computation time, we analyze in this paper the Schwarz waveform relaxation algorithm with Robin transmission conditions. The Robin transmission conditions contain a free parameter, which has a significant effect on the convergence rate of the Schwarz waveform relaxation algorithm. Determining the Robin parameter is therefore one of the top-priority matters for the study of the Schwarz waveform relaxation algorithm. We provide new formula to fix the Robin parameter and we show numerically that the new Robin parameter is more efficient than the one proposed previously in the literature.

Sign in / Sign up

Export Citation Format

Share Document