Numerical Errors in Unsteady Flow Simulations

2019 ◽  
Vol 4 (2) ◽  
Author(s):  
L. Eça ◽  
G. Vaz ◽  
S. L. Toxopeus ◽  
M. Hoekstra
Keyword(s):  
Statistical Convergence ◽  
Initial Condition ◽  
Convergence Criteria ◽  
Courant Number ◽  
Time Step ◽  
Flow Simulations ◽  
Numerical Errors ◽  
Data Points ◽  
Discretization Errors ◽  
Iterative Error

This article discusses numerical errors in unsteady flow simulations, which may include round-off, statistical, iterative, and time and space discretization errors. The estimation of iterative and discretization errors and the influence of the initial condition on unsteady flows that become periodic are discussed. In this latter case, the goal is to determine the simulation time required to reduce the influence of the initial condition to negligible levels. Two one-dimensional, unsteady manufactured solutions are used to illustrate the interference between the different types of numerical errors. One solution is periodic and the other includes a transient region before it reaches a steady-state. The results show that for a selected grid and time-step, statistical convergence of the periodic solution may be achieved at significant lower error levels than those of iterative and discretization errors. However, statistical convergence deteriorates when iterative convergence criteria become less demanding, grids are refined, and Courant number increased.For statistically converged solutions of the periodic flow and for the transient solution, iterative convergence criteria required to obtain a negligible influence of the iterative error when compared to the discretization error are more strict than typical values found in the open literature. More demanding criteria are required when the grid is refined and/or the Courant number is increased. When the numerical error is dominated by the iterative error, it is pointless to refine the grid and/or reduce the time-step. For solutions with a numerical error dominated by the discretization error, three different techniques are applied to illustrate how the discretization uncertainty can be estimated, using grid/time refinement studies: three data points at a fixed Courant number; five data points involving three time steps for the same grid and three grids for the same time-step; five data points including at least two grids and two time steps. The latter two techniques distinguish between space and time convergence, whereas the first one combines the effect of the two discretization errors.

scholarly journals Overview of the 2018 Workshop on Iterative Errors in Unsteady Flow Simulations

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
L. Eça ◽  
G. Vaz ◽  
M. Hoekstra ◽  
S. Pal ◽  
E. Muller ◽  
...  
Keyword(s):  
Unsteady Flow ◽  
Time Integration ◽  
Implicit Time ◽  
Test Case ◽  
Convergence Criteria ◽  
Courant Number ◽  
Time Step ◽  
Flow Solver ◽  
Flow Simulations ◽  
Implicit Time Integration

Abstract Two workshops were held at the ASME V&V Symposiums of 2017 and 2018 dedicated to Iterative Errors in Unsteady Flow Simulations. The focus was on the effect of iterative errors on numerical simulations performed with implicit time integration, which require the solution of a nonlinear set of equations at each time-step. The main goal of these workshops was to create awareness to the problem and to confirm that different flow solvers exhibited the same trends. The test case was a simple two-dimensional, laminar flow of a single-phase, incompressible, Newtonian fluid around a circular cylinder at the Reynolds number of 100. A set of geometrically similar multiblock structured grids was available and boundary conditions to perform the simulations were proposed to the participants. Results from seven flow solvers were submitted, but not all of them followed exactly the proposed conditions. One set of results was obtained with adaptive grid and time refinement using triangular elements (CADYF) and another used a compressible flow solver with a dual time stepping technique and a Mach number of 0.2 (DLR-Tau). The remaining five submissions were obtained with five different incompressible flow solvers (ansyscfx 14.5, pimplefoam, refresco, saturne, starccm+ v12.06.010-R8) using implicit time integration in the proposed grids. The results obtained in this simple test case showed that iterative errors may have a significant impact on the numerical accuracy of unsteady flow simulations performed with implicit time integration. Iterative errors can be significantly larger (one to two orders of magnitude) than the residuals and/or solution changes used as convergence criteria at each time-step. The Courant number affected the magnitude of the iterative errors obtained in the proposed exercise. For the same iterative convergence criteria at each time-step, increasing the Courant number tends to increase the iterative error.

scholarly journals A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

2019 ◽  
Vol 19 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Max Gunzburger ◽  
Nan Jiang ◽  
Zhu Wang
Keyword(s):  
Stokes Equations ◽  
Second Order ◽  
Navier Stokes ◽  
Conditional Stability ◽  
Initial Condition ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Flow Simulations ◽  
Flow Problems ◽  
Time Stepping

AbstractWe consider settings for which one needs to perform multiple flow simulations based on the Navier–Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

Code Verification of ReFRESCO With a Statistically Periodic Manufactured Solution

2014 ◽  
Author(s):  
Luís Eça ◽  
Guilherme Vaz ◽  
Martin Hoekstra
Keyword(s):  
Error Estimation ◽  
Unsteady Flows ◽  
Fixed Time ◽  
Convergence Criteria ◽  
Code Verification ◽  
Time Step ◽  
Space And Time ◽  
Manufactured Solution ◽  
Discretization Errors

This paper presents a Code Verification study performed with the unsteady ensemble-averaged Navier-Stokes (URANS) solver ReFRESCO using the Method of Manufactured Solutions. The study uses a statistically periodic manufactured solution including the un-damped eddy-viscosity of the Spalart & Allmaras turbulence model. Three main aspects of the numerical calculations of unsteady flows are addressed in this study: iterative errors; discretization errors (space and time) and the determination of the observed order of (space and time) convergence. The availability of an exact solution allows the determination of the numerical error and so the effects of iterative and discretization errors can be addressed. The paper presents grid and time refinement studies with different (iterative) convergence criteria and demonstrates that grid and time resolution are strongly connected when attempts are made to minimize the numerical uncertainty in the calculation of unsteady flows. The paper also addresses error estimation based on power series expansions in the calculation of unsteady (space and time dependent) flows. Simultaneous grid and time refinement is compared to grid refinement with fixed time step and time refinement with fixed grid. The advantages and limitations of both options are discussed in the context of Code Verification (error evaluation) and Solution Verification (error estimation).

On the Friction Coefficient in the Numerical Simulation of Tidal Wave Propagation

2010 ◽  
Author(s):  
Z. Y. Song ◽  
C. Cheng ◽  
F. M. Xu ◽  
J. Kong
Keyword(s):  
Numerical Simulation ◽  
Wave Propagation ◽  
Friction Factor ◽  
Tidal Wave ◽  
Computational Time ◽  
Numerical Dissipation ◽  
Courant Number ◽  
Time Step ◽  
Large Time Step ◽  
The Relationship

Based on the analytical solution of one-dimensional simplified equation of damping tidal wave and Heuristic stability analysis, the precision of numerical solution, computational time and the relationship between the numerical dissipation and the friction dissipation are discussed with different numerical schemes in this paper. The results show that (1) when Courant number is less than unity, the explicit solution of tidal wave propagation has higher precision and requires less computational time than the implicit one; (2) large time step is allowed in the implicit scheme in order to reduce the computational time, but the precision of the solution also reduce and the calculation precision should be guaranteed by reducing the friction factor: (3) the friction factor in the implicit solution is related to Courant number, presented as the determined friction factor is smaller than the natural value when Courant number is larger than unity, and their relationship formula is given from the theoretical analysis and the numerical experiments. These results have important application value for the numerical simulation of the tidal wave.

scholarly journals Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

2014 ◽  
Vol 33 (2) ◽  
pp. 59-67
Author(s):  
Pankaj Kumar ◽  
S. S. Bhatia ◽  
Vijay Kumar
Keyword(s):  
Statistical Convergence ◽  
Convergence Criteria ◽  
Double Sequences ◽  
Normed Spaces ◽  
Real Numbers ◽  
Positive Real ◽  
Probabilistic Normed Spaces

In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers $\lambda=(\lambda_{n})$ and $\mu = (\mu_{n})$  such that each tending to zero, also $\lambda_{n+1}\leq \lambda_{n}+1, \lambda_{1}=1,$ and $\mu_{n+1}\leq \mu_{n}+1, \mu_{1}=1.$ We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces.

scholarly journals An orthogonal terrain-following coordinate and its preliminary tests using 2-D idealized advection experiments

2014 ◽  
Vol 7 (4) ◽  
pp. 1767-1778 ◽  
Author(s):  
Y. Li ◽  
B. Wang ◽  
D. Wang ◽  
J. Li ◽  
L. Dong
Keyword(s):  
Rotation Angle ◽  
Computational Grids ◽  
Time Step ◽  
Numerical Errors ◽  
Terrain Following ◽  
Steep Terrain ◽  
Definition Of ◽  
Σ Coordinate ◽  
Basis Vectors ◽  
Better Than

Abstract. We have designed an orthogonal curvilinear terrain-following coordinate (the orthogonal σ coordinate, or the OS coordinate) to reduce the advection errors in the classic σ coordinate. First, we rotate the basis vectors of the z coordinate in a specific way in order to obtain the orthogonal, terrain-following basis vectors of the OS coordinate, and then add a rotation parameter b to each rotation angle to create the smoother vertical levels of the OS coordinate with increasing height. Second, we solve the corresponding definition of each OS coordinate through its basis vectors; and then solve the 3-D coordinate surfaces of the OS coordinate numerically, therefore the computational grids created by the OS coordinate are not exactly orthogonal and its orthogonality is dependent on the accuracy of a numerical method. Third, through choosing a proper b, we can significantly smooth the vertical levels of the OS coordinate over a steep terrain, and, more importantly, we can create the orthogonal, terrain-following computational grids in the vertical through the orthogonal basis vectors of the OS coordinate, which can reduce the advection errors better than the corresponding hybrid σ coordinate. However, the convergence of the grid lines in the OS coordinate over orography restricts the time step and increases the numerical errors. We demonstrate the advantages and the drawbacks of the OS coordinate relative to the hybrid σ coordinate using two sets of 2-D linear advection experiments.

scholarly journals Effects of spatial discretization in ice-sheet modelling using the shallow-ice approximation

2006 ◽  
Vol 52 (176) ◽  
pp. 89-98 ◽  
Author(s):  
J. Van Den Berg ◽  
R.S.W. Van De Wal ◽  
J. Oerlemans
Keyword(s):  
Initial Conditions ◽  
Strong Dependence ◽  
Ice Sheet ◽  
Model Parameters ◽  
Time Step ◽  
Numerical Errors ◽  
Surface Gradient ◽  
Dimension Analysis ◽  
Sloping Bed ◽  
Ice Model

AbstractThis paper assesses a two-dimensional, vertically integrated ice model for its numerical properties in the calculation of ice-sheet evolution on a sloping bed using the shallow-ice approximation. We discuss the influence of initial conditions and individual model parameters on the model’s numerical behaviour, with emphasis on varying spatial discretizations. The modelling results suffer badly from numerical problems. They show a strong dependence on gridcell size and we conclude that the widely used gridcell spacing of 20 km is too coarse. The numerical errors are small in each single time-step, but increase non-linearly over time and with volume change, as a result of feedback of the mass balance with height. We propose a new method for the calculation of the surface gradient near the margin, which improves the results significantly. Furthermore, we show that we may use dimension analysis as a tool to explain in which situations numerical problems are to be expected.

Using Graphics Processing Units to Accelerate Numerical Simulations of Interfacial Incompressible Flows

2012 ◽  
Author(s):  
Stephen R. Codyer ◽  
Mehdi Raessi ◽  
Gaurav Khanna
Keyword(s):  
Linear Algebra ◽  
Graphics Processing Units ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Surface Tension Force ◽  
Double Precision ◽  
Convergence Criteria ◽  
Time Step ◽  
Two Phase ◽  
Poisson Problem

We present a GPU accelerated numerical solver for incompressible, immiscible, two-phase fluid flows. This leads to a significant simulation speed-up and thus, the capability to have finer grid sizes and/or more accurate convergence criteria. We solve the Navier-Stokes equations, which include the surface tension force, by using a two-step projection method requiring the iterative solution to a pressure Poisson problem at each time step. However, running a serial linear algebra solver on a CPU to solve the pressure Poisson problem can take 50–99.9% of the total simulation time. To remove this bottleneck, we employ the large parallelization capabilities of GPUs by developing a double-precision parallel linear algebra solver, SCGPU, using NVIDIA’s CUDA v.4.0 libraries. The performance of SCGPU in serial simulations is presented, in addition to an evaluation of two pre-packaged GPU linear algebra solvers CUSP and CULA-sparse. We also present preliminary results of a GPU-accelerated MPI CPU flow solver.

Multipoint Explicit Differencing (MED) for Time Integrations of the Wave Equation

2007 ◽  
Vol 135 (11) ◽  
pp. 3862-3875 ◽  
Author(s):  
Celal S. Konor ◽  
Akio Arakawa
Keyword(s):  
Wave Equation ◽  
Finite Number ◽  
Multilevel Models ◽  
Horizontal Resolution ◽  
Explicit Scheme ◽  
Implicit Scheme ◽  
Courant Number ◽  
Time Step ◽  
Wide Range ◽  
Grid Points

Abstract For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

scholarly journals Acceleration of the IMplicit–EXplicit nonhydrostatic unified model of the atmosphere on manycore processors

2017 ◽  
Vol 33 (2) ◽  
pp. 242-267 ◽  
Author(s):  
Daniel S Abdi ◽  
Francis X Giraldo ◽  
Emil M Constantinescu ◽  
Lester E Carr ◽  
Lucas C Wilcox ◽  
...  
Keyword(s):  
Weather Prediction ◽  
Time Integration ◽  
Explicit Methods ◽  
Benchmark Problems ◽  
Graphic Processing Units ◽  
Courant Number ◽  
Time Step ◽  
Integration Methods ◽  
Manycore Processors ◽  
Time Integration Methods

We present the acceleration of an IMplicit–EXplicit (IMEX) nonhydrostatic atmospheric model on manycore processors such as graphic processing units (GPUs) and Intel’s Many Integrated Core (MIC) architecture. IMEX time integration methods sidestep the constraint imposed by the Courant–Friedrichs–Lewy condition on explicit methods through corrective implicit solves within each time step. In this work, we implement and evaluate the performance of IMEX on manycore processors relative to explicit methods. Using 3D-IMEX at Courant number C = 15, we obtained a speedup of about 4× relative to an explicit time stepping method run with the maximum allowable C = 1. Moreover, the unconditional stability of IMEX with respect to the fast waves means the speedup can increase significantly with the Courant number as long as the accuracy of the resulting solution is acceptable. We show a speedup of 100× at C = 150 using 1D-IMEX to demonstrate this point. Several improvements on the IMEX procedure were necessary in order to outperform our results with explicit methods: (a) reducing the number of degrees of freedom of the IMEX formulation by forming the Schur complement, (b) formulating a horizontally explicit vertically implicit 1D-IMEX scheme that has a lower workload and better scalability than 3D-IMEX, (c) using high-order polynomial preconditioners to reduce the condition number of the resulting system, and (d) using a direct solver for the 1D-IMEX method by performing and storing LU factorizations once to obtain a constant cost for any Courant number. Without all of these improvements, explicit time integration methods turned out to be difficult to beat. We discuss in detail the IMEX infrastructure required for formulating and implementing efficient methods on manycore processors. Several parametric studies are conducted to demonstrate the gain from each of the abovementioned improvements. Finally, we validate our results with standard benchmark problems in numerical weather prediction and evaluate the performance and scalability of the IMEX method using up to 4192 GPUs and 16 Knights Landing processors.

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