scholarly journals IMPROVING A POSTERIORI SPATIAL DISCRETIZATION ERROR ESTIMATION FOR SN TRANSPORT BY SINGULAR CHARACTERISTICS TRACKING

2021 ◽  
Vol 247 ◽  
pp. 03022
Author(s):  
Nathan H. Hart ◽  
Yousry Y. Azmy
Keyword(s):  
Error Estimate ◽  
Numerical Solution ◽  
Computational Cost ◽  
Discretization Error ◽  
Error Estimator ◽  
Approximation Procedure ◽  
Spatial Discretization ◽  
True Solution ◽  
Error Estimators ◽  
Singular Characteristics

Previously, we have developed a novel spatial discretization error estimator, the “residual source” estimator, in which an error transport problem, analogous to the discretized transport equation, is solved to acquire an estimate of the error, with a residual term acting as a fixed source. Like all error estimators, the residual source estimator suffers inaccuracy and imprecision in the proximity of singular characteristics, lines across which the solution is irregular. Estimator performance worsens as the irregularities become more pronounced, especially so if the true solution itself is discontinuous. This work introduces a modification to the residual approximation procedure that seeks to reduce the adverse effects of the singular characteristics on the error estimate. A partial singular characteristic tracking scheme is implemented to reduce the portion of the error in the numerical solution born by irregularities in the true solution. This treated numerical solution informs the residual approximations. The partial singular characteristic tracking scheme greatly enhances the numerical solution for a problem with prominent singular characteristics. The residual approximation and resultant residual source error estimate are likewise improved by the scheme, which only incurs the computational cost of an extra inner iteration.

scholarly journals Application of discretization error estimators in stepped column buckling problems

2021 ◽  
Vol 37 (1) ◽  
Author(s):  
B. Souza ◽  
D. Fernades ◽  
C. Anflor ◽  
M. Morais
Keyword(s):  
Cross Section ◽  
Discretization Error ◽  
Error Estimator ◽  
Column Buckling ◽  
Error Estimators ◽  
Convergence Error ◽  
Solution Convergence ◽  
Richardson’S Extrapolation ◽  
The Finite Difference Method ◽  
Convergent Solution

In order to reduce the discretization error, in this paper, Richardson’s Extrapolation and Convergence Error Estimator were used to investigating the buckling problem convergence. The main objective was to verify the convergence order of the stepped column problem and to define a consistent moment of inertia at the point of variation of the cross-section. The variable of interest was the critical buckling load obtained by the Finite Difference Method. The convergent solution obtained errors less than 10-8, and this work showed that the best solution is not defined by excessive mesh refinement, but by the solution convergence analysis.

scholarly journals AN EFFICIENT AND ROBUST GPGPU-BASED SHALLOW WATER MODEL

2020 ◽  
pp. 47
Author(s):  
Fatima-zahra Mihami ◽  
Volker Roeber
Keyword(s):  
Numerical Solution ◽  
Shallow Water ◽  
Computational Cost ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Water Model ◽  
Staggered Grid ◽  
Finite Volume Scheme ◽  
Flow Solver ◽  
Riemann Solvers

We present an efficient and robust numerical model for the solution of the Shallow Water Equations with the objective to develop the numerical foundation for an advanced free surface flow solver. The numerical solution is based on an explicit Finite Volume scheme on a staggered grid to ensure the conservation of mass and momentum across flow discontinuities and wet-dry transitions. This leads to an accurate numerical solution at low computational cost without the need for Riemann solvers. The efficiency of the lean numerical structure is further optimized through a CUDA-GPU implementation.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/xMnK_r7Tj1Q

scholarly journals Error Estimate and Adaptive Refinement in Mixed Discrete Least Squares Meshless Method

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
J. Amani ◽  
A. Saboor Bagherzadeh ◽  
T. Rabczuk
Keyword(s):  
Convergence Rate ◽  
Error Estimate ◽  
Least Squares ◽  
Meshless Method ◽  
Least Square ◽  
Adaptive Refinement ◽  
Error Estimator ◽  
Shape Functions ◽  
Numerical Errors ◽  
Elasticity Problems

The node moving and multistage node enrichment adaptive refinement procedures are extended in mixed discrete least squares meshless (MDLSM) method for efficient analysis of elasticity problems. In the formulation of MDLSM method, mixed formulation is accepted to avoid second-order differentiation of shape functions and to obtain displacements and stresses simultaneously. In the refinement procedures, a robust error estimator based on the value of the least square residuals functional of the governing differential equations and its boundaries at nodal points is used which is inherently available from the MDLSM formulation and can efficiently identify the zones with higher numerical errors. The results are compared with the refinement procedures in the irreducible formulation of discrete least squares meshless (DLSM) method and show the accuracy and efficiency of the proposed procedures. Also, the comparison of the error norms and convergence rate show the fidelity of the proposed adaptive refinement procedures in the MDLSM method.

scholarly journals Implementation and performance of adaptive mesh refinement in the Ice Sheet System Model (ISSM v4.14)

2019 ◽  
Vol 12 (1) ◽  
pp. 215-232 ◽  
Author(s):  
Thiago Dias dos Santos ◽  
Mathieu Morlighem ◽  
Hélène Seroussi ◽  
Philippe Remy Bernard Devloo ◽  
Jefferson Cardia Simões
Keyword(s):  
Adaptive Mesh Refinement ◽  
Mesh Refinement ◽  
Computational Cost ◽  
Ice Sheet ◽  
Adaptive Mesh ◽  
System Model ◽  
Computational Time ◽  
Error Estimator ◽  
Fine Mesh ◽  
Grounding Line

Abstract. Accurate projections of the evolution of ice sheets in a changing climate require a fine mesh/grid resolution in ice sheet models to correctly capture fundamental physical processes, such as the evolution of the grounding line, the region where grounded ice starts to float. The evolution of the grounding line indeed plays a major role in ice sheet dynamics, as it is a fundamental control on marine ice sheet stability. Numerical modeling of a grounding line requires significant computational resources since the accuracy of its position depends on grid or mesh resolution. A technique that improves accuracy with reduced computational cost is the adaptive mesh refinement (AMR) approach. We present here the implementation of the AMR technique in the finite element Ice Sheet System Model (ISSM) to simulate grounding line dynamics under two different benchmarks: MISMIP3d and MISMIP+. We test different refinement criteria: (a) distance around the grounding line, (b) a posteriori error estimator, the Zienkiewicz–Zhu (ZZ) error estimator, and (c) different combinations of (a) and (b). In both benchmarks, the ZZ error estimator presents high values around the grounding line. In the MISMIP+ setup, this estimator also presents high values in the grounded part of the ice sheet, following the complex shape of the bedrock geometry. The ZZ estimator helps guide the refinement procedure such that AMR performance is improved. Our results show that computational time with AMR depends on the required accuracy, but in all cases, it is significantly shorter than for uniformly refined meshes. We conclude that AMR without an associated error estimator should be avoided, especially for real glaciers that have a complex bed geometry.

scholarly journals Multilevel weighted least squares polynomial approximation

2020 ◽  
Vol 54 (2) ◽  
pp. 649-677 ◽  
Author(s):  
Abdul-Lateef Haji-Ali ◽  
Fabio Nobile ◽  
Raúl Tempone ◽  
Sören Wolfers
Keyword(s):  
Least Squares ◽  
Polynomial Approximation ◽  
Weighted Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Discretization Error ◽  
Logarithmic Factor ◽  
Complexity Bounds ◽  
Single Level ◽  
Random Samples

Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

scholarly journals A New Algorithm for Solving Terminal Value Problems of q-Difference Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Yong-Hong Fan ◽  
Lin-Lin Wang
Keyword(s):  
Exact Solution ◽  
Numerical Methods ◽  
Error Estimate ◽  
Numerical Solution ◽  
Numerical Method ◽  
Difference Equations ◽  
Initial Value Problems ◽  
Convergence Speed ◽  
Initial Value ◽  
Delta Derivatives

We propose a new algorithm for solving the terminal value problems on a q-difference equations. Through some transformations, the terminal value problems which contain the first- and second-order delta-derivatives have been changed into the corresponding initial value problems; then with the help of the methods developed by Liu and H. Jafari, the numerical solution has been obtained and the error estimate has also been considered for the terminal value problems. Some examples are given to illustrate the accuracy of the numerical methods we proposed. By comparing the exact solution with the numerical solution, we find that the convergence speed of this numerical method is very fast.

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