A Space-Time Parallel Method to Solve Space-Dependent Neutron Kinetics Equations in Hexagonal-Z Geometry

2018 ◽  
Author(s):  
Zhizhu Zhang ◽  
Yun Cai ◽  
Xingjie Peng ◽  
Qing Li
Keyword(s):  
Nuclear Reactor ◽  
Time Integration ◽  
Euler Method ◽  
Space Time ◽  
Parallel Method ◽  
Sequential Method ◽  
Neutron Kinetics ◽  
Nodal Method ◽  
Backward Euler ◽  
Parareal Method

Neutron kinetics plays an important role in reactor safety and analysis. The backward Euler method is the most widely used time integration method in the calculation of space-dependent nuclear reactor kinetics. Diagonally Implicit Runge-Kutta (DIRK) method owns high accuracy and excellent stability and it could be applied to the neutron kinetics for hexagonal-z geometry application. As solving the neutron kinetics equations is very time-consuming and the number of available cores continues to increase with parallel architectures evolving, parallel algorithms need to be designed to utilize the available resources effectively. However, it is difficult to parallel in time axis since the later moment is strongly dependent on the previous moment. In this paper, the Parareal method which is a time parallel method and implemented by MPI in the processor level is studied in the hexagonal-z geometry with the help of DIRK method. In order to make good use of the parallelism, a parallel strategy in the space direction is also used. In the coarse nodal method, many same operations are finished in the nodes and these operations could be parallel by OpenMP in the thread level since they are independent. Several transient cases are used to validate this method. The results show that the Parareal method gets a fast-convergent speed such as only 2∼3 iterations are needed to convergent. This space-time parallel method could reduce the cost time compared to the sequential method.

A Comparative Study of Ten Different Methods on Numerical Solving of Point Reactor Neutron Kinetics Equations

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Yining Zhang ◽  
Haochun Zhang ◽  
Kexin Wang
Keyword(s):  
Nuclear Power ◽  
Nuclear Reactor ◽  
Reactor Core ◽  
Density Variation ◽  
Euler Method ◽  
Reactor Neutron ◽  
Computational Time ◽  
Neutron Density ◽  
Power Series Method ◽  
Neutron Kinetics

Point reactor neutron kinetics equations describe the time-dependent neutron density variation in a nuclear reactor core. These equations are widely applied to nuclear system numerical simulation and nuclear power plant operational control. This paper analyzes the characteristics of ten different basic or normal methods to solve the point reactor neutron kinetics equations. The accuracy after introducing different kinds of reactivity, stiffness of methods, and computational efficiency are analyzed. The calculation results show that: considering both the accuracy and stiffness, implicit Runge–Kutta method and Hermite method are more suitable for solution on these given conditions. The explicit Euler method is the fastest, while the power series method spends the most computational time.

scholarly journals Composite Backward Differentiation Formula for the Bidomain Equations

2020 ◽  
Vol 11 ◽  
Author(s):  
Xindan Gao ◽  
Craig S. Henriquez ◽  
Wenjun Ying
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Euler Method ◽  
Implicit Methods ◽  
Differentiation Formula ◽  
Bidomain Equations ◽  
Time Integration Method ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Backward Euler

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.

COMPUTATIONAL DYNAMICS OF ORDINARY DIFFERENTIAL EQUATIONS

1995 ◽  
Vol 05 (01) ◽  
pp. 159-174 ◽  
Author(s):  
MAXIM POLIASHENKO ◽  
CYRUS K. AIDUN
Keyword(s):  
Time Integration ◽  
Euler Method ◽  
Point Of View ◽  
Implicit Time ◽  
Time Step ◽  
Navier Stokes Equation ◽  
True Solution ◽  
Integration Schemes ◽  
Backward Euler ◽  
Chaotic Solution

Discrete schemes, used to perform time integration of ODE’s, are expected to exhibit qualitatively ‘true’ dynamics in terms of the solutions and their stability. In past years, it has been discovered that such discretizations may cause spurious steady states and some explicit schemes may produce ‘computational chaos.’ In this study, we show that implicit time integration schemes, such as the backward Euler method, can also produce computationally chaotic solutions. Furthermore, we show that the opposite phenomenon may also take place both for explicit and for implicit schemes: computationally generated ‘spurious order’ may replace the true chaotic solution before the scheme becomes linearly unstable. The numerical solution may become chaotic again as the discretization step is further increased. The spurious computational order and chaos are discussed by solving low-dimensional dynamical systems, as well as a large system of ODE representing the solution to the Navier-Stokes equation. Our results support the point of view that the deviations in the behavior of the computed solution from the true solution has deterministic character with the time step playing the role of an artificial bifurcation parameter.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

scholarly journals A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 769-782
Author(s):  
Amiya K. Pani ◽  
Vidar Thomée ◽  
A. S. Vasudeva Murthy
Keyword(s):  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Second Order ◽  
Time Step ◽  
Convection Diffusion ◽  
First Order ◽  
Backward Euler ◽  
Strang Splitting ◽  
Spatially Periodic

AbstractWe analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

scholarly journals FDTD Computation of Space/Time Integrated Electromagnetic Lagrangian: New Insights on Mutually Coupled Wide-band Antenna Design

2021 ◽  
Author(s):  
Debdeep Sarkar ◽  
Yahia Antar
Keyword(s):  
Electromagnetic Interference ◽  
Interference Mitigation ◽  
Near Field ◽  
Time Integration ◽  
Wide Band ◽  
Space Time ◽  
Ultra Wideband ◽  
Mimo Antennas ◽  
Electric And Magnetic Fields ◽  
Spatio Temporal

In this paper, we develop a formalism based on either spatially or temporally integrated electromagnetic (EM) Lagrangian, which provides new insights about the near-field reactive energy around generic antennas for arbitrary spatio-temporal excitation signals. Using electric and magnetic fields calculated via FDTD technique and interpolation routines, we compute and plot the normalized values of space/time integrated EM Lagrangian around antennas. While the time-integration of EM Lagrangian sheds light onto the spatial distribution of inductive/capacitive reactive energy, time-variation of spatially integrated EM Lagrangian can help in design of ultra-wideband (UWB) MIMO antennas with low mutual coupling. The EM Lagrangian approach can assist in design of energy harvesting and wireless power transfer systems, as well as for electromagnetic interference mitigation applications.

Depletion Calculation for a Nodal Reactor Physics Code

2010 ◽  
Author(s):  
Antonio Carlos Marques Alvim ◽  
Fernando Carvalho da Silva ◽  
Aquilino Senra Martinez
Keyword(s):  
Diffusion Equation ◽  
Nuclear Reactor ◽  
Reactor Core ◽  
Expansion Method ◽  
Design System ◽  
Pressurized Water Reactor ◽  
Reactor Physics ◽  
Pressurized Water ◽  
Nodal Method ◽  
Orthogonal Polynomial Expansion

This paper deals with an alternative numerical method for calculating depletion and production chains of the main isotopes found in a pressurized water reactor. It is based on the use of the exponentiation procedure coupled to orthogonal polynomial expansion to compute the transition matrix associated with the solution of the differential equations describing isotope concentrations in the nuclear reactor. Actually, the method was implemented in an automated nuclear reactor core design system that uses a quick and accurate 3D nodal method, the Nodal Expansion Method (NEM), aiming at solving the diffusion equation describing the spatial neutron distribution in the reactor. This computational system, besides solving the diffusion equation, also solves the depletion equations governing the gradual changes in material compositions of the core due to fuel depletion. The depletion calculation is the most time-consuming aspect of the nuclear reactor design code, and has to be done in a very precise way in order to obtain a correct evaluation of the economic performance of the nuclear reactor. In this sense, the proposed method was applied to estimate the critical boron concentration at the end of the cycle. Results were compared to measured values and confirm the effectiveness of the method for practical purposes.

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