A High-Order, Time-Dependent Response Matrix Method for Reactor Kinetics

Until recently, reactor transient problems were exclusively solved by approximate deterministic methods. The increase in available computing power made it feasible to approach the transient analyses with time-dependent Monte Carlo methods. These methods offer the first-principle solution to the space-time evolution of reactor power by explicitly tracking prompt neutrons, precursors of delayed neutrons and delayed neutrons in time and space. Nevertheless, a very significant computing cost is associated with such methods. The general benefits of the Monte Carlo approach may be retained at a reduced computing cost by applying a hybrid stochastic-deterministic computing scheme. Among such schemes are those based on the fission matrix and the response matrix formalisms. These schemes aim at estimating a variant of the Greens function during a Monte Carlo transport calculation, which is later used to formulate a deterministic approach to solving a space-time dependent problem. In this contribution, we provide an overview of the time-dependent response matrix method, which describes a system by a set of response functions. We have recently suggested an approach where the functions are determined during a Monte Carlo criticality calculation and are then used to deterministically solve the space-time behaviour of the system. Here, we compare the time-dependent response matrix solution with the transient fission matrix and the time-dependent Monte Carlo solutions for a control rod movement problem in a mini-core reactor geometry. The response matrix formalism results in a set of loosely connected equations which offers favourable scaling properties compared to the methods based on the fission matrix formalism.

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Benchmarking with the multigroup diffusion high-order response matrix method

Annals of Nuclear Energy ◽

10.1016/0306-4549(91)90098-i ◽

1991 ◽

Vol 18 (9) ◽

pp. 535-544 ◽

Cited By ~ 19

Author(s):

E.Z. Müller ◽

Z.J. Weiss

Keyword(s):

Matrix Method ◽

High Order ◽

Response Matrix

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Tailoring High Order Time Discretizations for Use with Spatial Discretizations of Hyperbolic PDEs

10.21236/ada627006 ◽

2015 ◽

Author(s):

Sigal Gottlieb

Keyword(s):

High Order ◽

Hyperbolic Pdes ◽

High Order Time

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Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽

10.3390/math9101113 ◽

2021 ◽

Vol 9 (10) ◽

pp. 1113

Author(s):

Isaías Alonso-Mallo ◽

Ana M. Portillo

Keyword(s):

Numerical Experiments ◽

Splitting Method ◽

Method Of Lines ◽

Time Integration ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

High Order ◽

Time Dependent ◽

Boundary Values ◽

Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

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High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

Communications on Applied Mathematics and Computation ◽

10.1007/s42967-020-00110-5 ◽

2021 ◽

Author(s):

Giacomo Albi ◽

Lorenzo Pareschi

Keyword(s):

Multistep Methods ◽

General Setting ◽

Reaction Diffusion ◽

Strong Stability ◽

High Order ◽

Iterative Solvers ◽

Time Dependent ◽

Linear Multistep Methods ◽

Advection Diffusion Equation ◽

Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

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