Backward Euler Finite Difference Research Articles

A Precision Benchmark Suite for Nuclear Reactor Point Kinetics Equations via Converged Accelerated Taylor Series (CATS)

Extreme benchmarks of 10 or more places for the point kinetics equations for time-dependent nuclear reactor power transients are rare. Therefore, to establish an extreme benchmark, we employ a Taylor series (TS) with continuous analytical continuation to solve the ordinary differential equations of point kinetics including feedback. Nonlinear Wynn-epsilon convergence acceleration confirms the highly precise solutions for neutron and precursor densities. Through adaptive partitioning of time intervals, the proposed Converged Accelerated Taylor Series, or CATS algorithm in double precision, automatically performs successive mesh refinement to obtain high-precision initial conditions for each subinterval, with the intent to reduce propagation error. Confirmation of 10 to 12 places comes from comparison to the BEFD (Backward Euler Finite Difference) algorithm in quadruple precision also developed by the author. We report benchmark results for common cases found in the literature including step, ramp, zigzag, and sinusoidal prescribed reactivity insertions and insertions with nonlinear adiabatic Doppler feedback. We also establish a suite of new prescribed reactivity insertions and insertions with feedback, based on reactivities with Taylor series representations as suggested by the CATS algorithm.

ALTERNATIVE DIRECTION IMPLICIT METHOD FOR SOLVING FIRST ORDER 2D HYPERBOLIC DELAY DIFFERENTIAL EQUATIONS

This article considers delayed two-dimensional first order hyperbolic differential equations. The propagation of the discontinuity of the solution is also established. An alternating implicit finite difference method and backward Euler finite difference methods are presented. We proved that this method is first-order convergent. Illustrating numerical examples are given for validation. We also present an application of the proposed approach to the variable delay problem.

Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Abstract We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) L^{\infty}_{t}(L^{\infty}_{x}) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ⁢ ( H x 1 ) L^{\infty}_{t}(H^{1}_{x}) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.

Arterial tissues and their inflammatory response to collagen damage: A continuum in silico model coupling nonlinear mechanics, molecular pathways, and cell behavior

Damage in soft biological tissues causes an inflammatory reaction that initiates a chain of events to repair the tissue. This work presents a continuum model and its in silico implementation that describe the cascade of mechanisms leading to tissue healing, coupling mechanical as well as chemo-biological processes. The mechanics is described by means of a Lagrangian nonlinear continuum mechanics framework and follows the homogenized constrained mixtures theory. Plastic-like damage, growth and remodeling as well as homeostasis are taken into account. The chemo-biological pathways account for two molecular and four cellular species, and are activated by damage of collagen molecules in fibers. To consider proliferation, differentiation, diffusion and chemotaxis of species, diffusion–advection–reaction equations are employed. To the best of authors’ knowledge, the proposed model combines for the first time such high number of chemo-mechano-biological mechanisms in a consistent continuum biomechanical framework. The resulting set of coupled differential equations describe balance of linear momentum, evolution of kinematic variables as well as mass balance equations. They are discretized in time according to a backward Euler finite difference scheme, and in space through a finite element Galerkin discretization. The features of the model are firstly demonstrated presenting the species dynamics and highlighting the influence of damage intensities on the growth outcome. In terms of a biaxial test, the chemo-mechano-biological coupling and the model’s applicability to reproduce normal as well as pathological healing are shown. A last numerical example underlines the model’s applicability to complex loading scenarios and inhomogeneous damage distributions. Concluding, the present work contributes towards comprehensive in silico models in biomechanics and mechanobiology.

A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels

This paper studies an accurate localized meshless collocation approach for solving two-dimensional nonlinear integro-differential equation (2D-NIDE) with multi-term kernels. The proposed strategy discretizes the unknown solution in two phases. First, the semi-discrete scheme is obtained by using backward Euler finite difference (FD) approach and the first-order convolution quadrature rule for the first order temporal derivative and the Riemann-Liouville (R-L) fractional integral, respectively. Second, the spatial discretization is established by means of the local radial basis function based on partition of unity (LRBF-PU) in the space variable and its partial derivatives. Furthermore, the unconditionally stable result and first-order convergence of the time semi-discrete scheme in L2-norm are proved by the energy method. It is shown that the proposed method is accurate and that the numerical results support the theoretical analysis.

Large eddy simulation of the wind flow in a realistic full-scale urban community with a scalable parallel algorithm

Turbulence and large computational domains make urban wind flow simulation a computationally challenging problem. To reduce the total simulation time and efficiently use today's new generation of supercomputers with massive processors, scalable parallel solvers are needed. In this work, we present a scalable domain decomposition method based 3D incompressible Navier-Stokes solver for the simulation of unsteady, complex wind flows around urban communities. A large eddy simulation with Smagorinsky modeling is employed for the simulation of the turbulent wind flows. On a fine unstructured grid, we discretize the spatial and temporal dimensions using a stabilized P1−P1 finite element method and an implicit backward Euler finite difference method, respectively. The resulting large-scale nonlinear algebraic system at each timestep is solved by a Newton-Krylov-Schwarz algorithm in parallel. The solver is first validated by a benchmark case where the numerical results are compared with measured data from a wind tunnel. Then, the wind flow field of a realistic actual-scale urban community with a group of buildings in the downtown area of Shenzhen, China, is studied. The numerical results show that the flow field matches with the experimental data for the benchmark problem, and some reasonable, detailed, and complex flow structures are obtained for the urban community simulation case. The scalability of the solver is studied on the TianHe-2A and Sunway TaihuLight supercomputers, and the solver scales up to 16,384 processor cores for a grid with over 20 million elements, which implies that the solver has the potential to perform fast and high-fidelity simulations of large-scale wind flows in complicated computational domains.

Comparison of Picard and Newton iterative schemes in the finite element modeling for one-dimensional variably saturated-unsaturated flows

Numerical simulation of saturated-unsaturated flow in permeable media is of incredible enthusiasm for some parts of science and engineering. Demonstrating of fluid flow in variably saturated-unsaturated permeable media generally brings about systems of highly nonlinear partial differential equations, Richards’ equation, which are not reasonable diagnostically except if ridiculous and distorting presumptions are made with respect to the attributes, flow dynamics, and properties of the physical frameworks. Picard and Newton methods are the most usually practiced iterative methods for the numerical solution of the nonlinearity coupled frameworks. A finite element numerical model was developed to precisely and efficiently figure a solution of the nonlinear Richards’ equation with homogeneous and layered soil. In this work, we have considered these two regular iterative strategies which can be utilized in a solution methodology for the nonlinear Richards' equation governing flow in variably saturated-unsaturated porous media. We have evaluated the efficiency, accuracy, robustness and computational efficacy of the iterative Newton and Picard methods. The assessments depend on three distinctive test issues of one-dimensional saturated-unsaturated flow with homogeneous and heterogeneous soil properties. Besides, spatial adjustment will be founded on a fine spatial discretization and temporal adjustment will be practiced utilizing variable order, variable step size dependent on the backward Euler finite difference formula. The computed outcomes obviously demonstrated that the methodology was all the more computationally productive, yet additionally increasingly precise and robust. Computational execution was significantly improved with the Picard strategy, and which could be utilized to simulate heterogeneous soil and Newton scheme is superior for homogeneous soil materials.

A uniformly convergent finite difference scheme for Robin type singularly perturbed parabolic convection diffusion problem

In this paper, a second order numerical scheme for solving a singularly perturbed convection diffusion problem with Robin boundary conditions is proposed. The numerical scheme is a combination of the fitted operator finite difference and the backward Euler finite difference methods. These are designed in order to solve respectively the spatial derivatives and the time derivative. Using some properties of the discrete problem the methods are analysed for convergence. Richardson extrapolation technique is used to improve the accuracy and also accelerate the convergence of the method. Numerical simulations are carried out to confirm the theoretical findings in the analysis before and after extrapolation.

Validation of the SHNC time-dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies

The diffusion approximation to the time-dependent Boltzmann transport equation gives accurate results for traditional nuclear reactor designs, but new reactor designs and new fuel elements require neutron transport methods. We develop a numerical approximation to the time-dependent transport equation coupled to delayed neutron precursors based on the spherical harmonics PL equations, for odd L, and on the Backward Euler finite difference discretization of time. The resulting scheme can be written as a stationary form of diffusive second order PL equations. This allows a reduction by half to the number of unknowns and also to apply a nodal collocation method to the spatial discretization of the problem, using coarse spatial grids to further reduce memory requirements. This scheme is validated with several transient benchmarks, where the convergence properties are established and compared with the simplified PL approximation. A more realistic transient benchmark, based on the two-group C5 MOX problem, is finally introduced, showing the need of high order PL approximation for complex fuel geometries.

Two-scale Finite Element Discretizations for Semilinear Parabolic Equations

In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝ^<i>d</i> with <i>d</i> = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying <i>H</i> = <i>O</i> (<i>h</i>^1/2), where <i>h</i> and <i>H</i> are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the <i>H</i>¹ (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from <i>O</i>(<i>h</i>^-<i>d</i>×<i>τ</i>^-1) to <i>O</i>(<i>h</i>^{-(<i>(d)</i>+1)/2}×<i>τ</i>^-1) where <i>τ</i> is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.

Modeling and simulation of cardiac electric activity in a human cardiac tissue with multiple ischemic zones.

In this work, a human ventricular model (ten Tusscher and Panfilov model) coupled with the tissue level monodomain model is used to analyze the influence of multiple myocardial ischemia on the human cardiac tissue. The existence and uniqueness of the ischemic model comprising the monodomain model with a discontinuous ionic model for the human cardiac tissue is discussed. The coupled system of partial differential equation and ordinary differential equations are solved numerically using [Formula: see text] finite elements in space and Backward Euler finite difference scheme in time. The apriori finite element error estimate for the numerical scheme has been shown to be of [Formula: see text]. Essentially, we evaluate the impact of the increasing size of the ischemic region and the presence of the multiple ischemic regions having equal or different intensities on the neighboring healthy part of the cardiac tissue. We examine both the individual and the combined influence of two types of ischemia, Hyperkalemia (with the variation of the extracellular potassium ion concentration, [Formula: see text]) and Hypoxia (with the variation of intracellular Adenosine triphosphate (ATP) concentration via parameter [Formula: see text]) on the cardiac electrical activity of cardiac tissue. We observe that with the increase in the ischemic region size by a factor five times, there is an additional almost 10% drop in the action potential duration (APD) in the neighboring healthy regions. The combined effect of Hyperkalemia and Hypoxia brings an additional 12% drop in APD in the ischemic subregions and an additional 5% drop in APD in the neighboring healthy part of the cardic tissue in comparison to the only Hyperkalemic ischemia. When the Hyperkalemic and/or Hypoxic degeneracy of a ischemic zone is non-uniform then innercore degeneracy has greater influence on resting potential and APD of outercore of variable intensity ischemic zone than the other way. Also, increasing the number of ischemic subregions from 2 to 4 leads to a 4% drop in APD.

Sharp error estimate for implicit finite element scheme for American put option

AbstractWe investigate a numerical solution method for a degenerate parabolic variational inequality that determines American vanilla put pricing. This method is based on piecewise linear finite elements in spatial variables and the backward Euler finite difference in time variable. For the approximate solution, we get sharp error estimate of orderO(h+τ3/4) in the energy norm of the corresponding differential operator.

Uniformly Convergent Second Order Numerical Method for a Class of Parameterized Singular Perturbation Problems

In this article, a class of nonlinear singularly perturbed boundary value problems depending on a parameter are considered. To solve this class of problems; first we apply the backward Euler finite difference scheme on Shishkin type meshes [standard Shishkin mesh (S-mesh), Bakhvalov–Shishkin mesh (B–S-mesh)]. The convergence analysis is carried out and the method is shown to be convergent with respect to the small parameter and is of almost first order accurate on S-mesh and first order accurate on B–S-mesh. Then, to improve the accuracy of the computed solution from almost first order to almost second order on S-mesh and from first order to second order on B–S-mesh, the post-processing method namely, the Richardson extrapolation technique is applied. The proof for the uniform convergence of the proposed method is carried out on both the meshes. Numerical experiments indicate the high accuracy of the proposed method.

A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid

AbstractA nonlinear singularly perturbed boundary value problem depending on a parameter is considered. First, we solve the problem using the backward Euler finite difference scheme on an adaptive grid. The adaptive grid is a special nonuniform mesh generated through equidistribution principle by a positive monitor function depending on the solution. The behavior of the solution, the stability and the error estimates are discussed. Then, the Richardson extrapolation technique is applied to improve the accuracy of the computed solution associated to the backward Euler scheme. The proofs of the uniform convergence for the backward Euler scheme and the Richardson extrapolation are carried out. Numerical experiments validate the theoretical estimates and indicates that the estimates are sharp.

Layer-adapted Meshes for Parameterized Singular Perturbation Problem

A nonlinear singularly perturbed boundary value problem depending on a parameter is considered. Two numerical methods are applied to solve this problem. First, we solve the problem using backward Euler finite difference scheme on layer adapted meshes. Then, Richardson extrapolation technique is applied to improve the accuracy of the computed solution. Numerical experiment is carried out to validate the theoretical estimates.

Kalman filters for assimilating near-surface observations into the Richards equation – Part 1: Retrieving state profiles with linear and nonlinear numerical schemes

Abstract. This paper examines the potential of different algorithms, based on the Kalman filtering approach, for assimilating near-surface observations into a one-dimensional Richards equation governing soil water flow in soil. Our specific objectives are: (i) to compare the efficiency of different Kalman filter algorithms in retrieving matric pressure head profiles when they are implemented with different numerical schemes of the Richards equation; (ii) to evaluate the performance of these algorithms when nonlinearities arise from the nonlinearity of the observation equation, i.e. when surface soil water content observations are assimilated to retrieve matric pressure head values. The study is based on a synthetic simulation of an evaporation process from a homogeneous soil column. Our first objective is achieved by implementing a Standard Kalman Filter (SKF) algorithm with both an explicit finite difference scheme (EX) and a Crank-Nicolson (CN) linear finite difference scheme of the Richards equation. The Unscented (UKF) and Ensemble Kalman Filters (EnKF) are applied to handle the nonlinearity of a backward Euler finite difference scheme. To accomplish the second objective, an analogous framework is applied, with the exception of replacing SKF with the Extended Kalman Filter (EKF) in combination with a CN numerical scheme, so as to handle the nonlinearity of the observation equation. While the EX scheme is computationally too inefficient to be implemented in an operational assimilation scheme, the retrieval algorithm implemented with a CN scheme is found to be computationally more feasible and accurate than those implemented with the backward Euler scheme, at least for the examined one-dimensional problem. The UKF appears to be as feasible as the EnKF when one has to handle nonlinear numerical schemes or additional nonlinearities arising from the observation equation, at least for systems of small dimensionality as the one examined in this study.

Microstructural Alterations in Bearing Steels under Rolling Contact Fatigue: Part 2—Diffusion-Based Modeling Approach

The microstructure of rolling element bearings can experience significant transformation when subjected to repetitive contact cycling. In Part 1 of this article, a detailed overview of the known microstructure alteration phenomena was presented and the different mechanisms for describing them proposed by various investigators were critically reviewed. It is agreed generally that the primary path of these structural changes is the decay of martensite due to carbon diffusion, leading to the formation of ferrite and lenticular carbides. Furthermore, these altered regions in the material microstructure can be stress concentration regions leading to crack initiation and propagation. In this article, a J2-based elastic–plastic Voronoi finite element model (previously developed by the authors) is coupled with a carbon diffusion–based model. Using Fick's law for stress-assisted diffusion, the dispersion of carbon in the bearing microstructure is evaluated. A backward Euler finite difference scheme is employed to solve the partial differential equations. The model can accurately predict the onset of martensitic decay and formation of the white etching bands along with their distinctive orientation. A comparison of the numerical results shows good corroboration with experimental observations.

Computational modeling of chemo-electro-mechanical coupling: a novel implicit monolithic finite element approach.

Computational modeling of the human heart allows us to predict how chemical, electrical, and mechanical fields interact throughout a cardiac cycle. Pharmacological treatment of cardiac disease has advanced significantly over the past decades, yet it remains unclear how the local biochemistry of an individual heart cell translates into global cardiac function. Here, we propose a novel, unified strategy to simulate excitable biological systems across three biological scales. To discretize the governing chemical, electrical, and mechanical equations in space, we propose a monolithic finite element scheme. We apply a highly efficient and inherently modular global-local split, in which the deformation and the transmembrane potential are introduced globally as nodal degrees of freedom, whereas the chemical state variables are treated locally as internal variables. To ensure unconditional algorithmic stability, we apply an implicit backward Euler finite difference scheme to discretize the resulting system in time. To increase algorithmic robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton-Raphson scheme. The proposed algorithm allows us to simulate the interaction of chemical, electrical, and mechanical fields during a representative cardiac cycle on a patient-specific geometry, robust and stable, with calculation times on the order of 4 days on a standard desktop computer.

The parameter uniform numerical method for singularly perturbed parabolic reaction–diffusion problems on equidistributed grids

This article presents the study of singularly perturbed parabolic reaction–diffusion problems with boundary layers. To solve these problems, we use a modified backward Euler finite difference scheme on layer adapted nonuniform meshes at each time level. The nonuniform meshes are obtained by equidistribution of a positive monitor function, which involves the second-order spatial derivative of the singular component of the solution. The equidistributing monitor function at each time level allows us to use this technique to non-linear parabolic problems. The truncation error and the stability analysis are obtained. Parameter–uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.

Analysis and Computation for Ground State Solutions of Bose--Fermi Mixtures at Zero Temperature

Previous numerical studies on the ground state structure of Bose--Fermi mixtures mostly relied on Thomas--Fermi (TF) approximation for the Fermi gas. In this paper, we establish the existence and uniqueness of ground state solutions of Bose--Fermi mixtures at zero temperature for both a coupled Gross--Pitaevskii (GP) equations model and a model with TF approximation for fermions. To prove the uniqueness, the key is to estimate the $L^\infty$ bounds of the ground state solution. By implementing an efficient method---gradient flow with discrete normalization with backward Euler finite difference discretization---to compute the coupled GP equations, we report extensive numerical results in one and two dimensions. The numerical experiments show that we can also extract many interesting phenomena without reference to TF approximation for the fermions. Finally, we numerically compare the ground state solutions for the coupled GP equations model and the model with TF approximation for fermions as well as for the model with TF approximations for both bosons and fermions.