Convergence in Numerical Science 12/11/2018 Kihwan Lee

Classification of Errors in Numerical Science - 1 ρ 𝜕𝑝 𝜕𝑛 = 𝑢2 𝑅 , where R = radius of curvature 𝜎max=𝜎 1+2 𝑎 𝑅 , where 2*a = crack length, R = radius of curvature Physical approximation error Physical modeling error: is the governing equation representative of the actual physics? Modeling approximation is always present. Good example: inviscid approximation, turbulence models Geometry modeling error: is the geometry good enough to capture the actual physics? Round and corners Is far-field boundary far enough? ~ 50 to 100 times of characteristic length Is boundary layer resolved enough? Is a round round-enough? Round corners introduces additional acceleration to the fluid (centrifugal). Is a sharp corner too sharp? Is the stress concentration too much? Computer round-off error Subtraction error Known problem in the computational science. Active research in the effort of transforming the governing equation to remove “subtraction” operation in the formulation Precision 32 bit computer: good up to 7 digit 64 bit computer: good up to 15 digit Considering round-off errors and subtraction errors, about 3 ~ 4 digits in 32-bit computer and about 5 ~ 6 digits in the 64-bit computer is trusted Iterative convergence error Solver error Direct vs. iterative solver Multi-grid vs. single grid solver Matrix behavior Condition number Importance of proper scaling – preconditioning matrix used often Discretization error Spatial discretization error How close is the derivative to the definition of “derivative”? Temporal discretization error Are all required time scale resolved properly to capture the propagation of physical information? Typical error estimate Error ~ O(Δ2), where 2nd order is accepted as standard. Programming error Higher level programming error – Fortran, C, C++ Lower level programming error – compiler, chip instruction System configuration error Library version mismatch User error Often biggest source of error Misinterpretation Lack of understanding of the governing physics

What is Convergence? Achievement of a limiting behavior in the solution of the governing equations and is typically represented by the diminishing residuals of the numerical solutions. It satisfies the governing equation used in the formulation Finite difference method Discretization on the differential form equation. Good for capturing differential quantities. Finite volume method Discretization on the integral form of equation. Good for captures conservative quantities. Finite element method Discretization on the weighted residual form of the integral equation. Introduces additional layer of approximation. Good for meshing. Generally non-conservative. It satisfies the applied discretization 1st order, 2nd order Upwind (using information from the upstream of the characteristic lines) and central difference Does it follow physical flow of information? Shock capturing has become successful only after taking this effect into consideration. Numerical damping Limiter: artificially introduced for the purpose of damping spurious solutions. Additional terms to satisfy convergence criteria. Precision How much precision to trust? About 5 ~ 6 digits from 64 bit computer, based on Navier Stokes solution. Mesh Boundary? How good is a good mesh? Often the given PDE is solved in the mesh generation process in order to minimize errors introduced by meshing alone.

Nature of Solution Structural Dynamics Typical solvers: Ansys, Abacus PDE: elliptic, hyperbolic Quantity of interest: Displacement: solver output Stress: derivative of the output Fluid Typical solvers: Ansys, Fluent, FloTherm PDE: hyperbolic, elliptic Quantity of interest: velocity, thermodynamic state, stress Velocity: solver output Thermodynamic state: solver output Electricity & Magnetism Typical solvers: HFSS E-fields, M-fields: derivative of the output Obtaining correct order of convergence on the derivative quantity often requires finer mesh compared to the obtaining convergence on the direct solver output. PDE Linear vs. Non-linear Coefficient independent/dependent of the solution. Elliptic Diffusion. Smooth solution. Diagonally dominant matrix. Better matrix convergence property. Parabolic Time-dependent diffusion. Similar to the elliptic case, but in transient. Hyperbolic Wave equation, convection equation. Sensitive to stability criteria. Requires finer resolution of time step to minimize numerical dissipation. Strong influence from the off-diagonal element. Poor matrix convergence property. Mixed direction of physical information flow at the boundaries. Heat Transfer Typical solvers: Sinda, Fluent, FloTherm PDE: parabolic, elliptic Quantity of interest: Temperature: solver output Heat flux: derivative of the output

Error Analysis Taylor series expansion Gradient x0 x0 + Δx x0 - Δx Gradient with 1st order approximation Taylor series expansion Expansion of a differentiable function 1st order form 2nd order formulation Accuracy of y(x0) depends on the relative size of Δ𝑥, y’’(x0), and y’’’(x0) Where y’’(x0) or y’’’(x0) is non-negligible, reduce Δ𝑥 Where y’’(x0) or y’’’(x0) is near zero, moderate size of Δ𝑥 is acceptable How large is acceptable? Gradient with 2nd order approximation 𝑦 𝑥0+Δ𝑥 =𝑦 𝑥0 + Δ𝑥 ∗ 𝑦 ′ 𝑥0 + Δ𝑥 2 2 𝑦 ′′ 𝑥0 + remainders 𝑦 ′ 𝑥0 = 𝑦 𝑥0+Δ𝑥 −𝑦 𝑥0 Δ𝑥 − Δ𝑥 2 𝑦 ′′ 𝑥0 𝑦 ′ 𝑥0 = 𝑦 𝑥0+Δ𝑥 −𝑦 𝑥0−Δ𝑥 2Δ𝑥 − Δ𝑥 2 6 𝑦 ′′′ 𝑥0

Convergence Criteria Analysis 𝜕𝑓 𝜕𝑡 +𝑢 𝜕𝑓 𝜕𝑥 −𝜇 𝜕2𝑓 𝜕𝑥2 =0 Convection-Diffusion Equation: Length scale to resolve both convection and diffusion terms is of the order of Convection term dominates in the region with larger cell length. Diffusion term dominates in the region with smaller cell length. Each convection and diffusion term require different max time scale. Boundary layer resolution dominates time step Combined effect of diffusion and convection for accuracy with 𝜇 = 10-5 and u = 1m/s, Presence of different time scale and length scales makes the problem hard to converge Multi-grid approach is often used for faster convergence – the difference is dramatic. Matrix pre-conditioning is often used for faster convergence. 𝐿= 𝜇 𝑢 𝜏𝑐𝑜𝑛𝑣≤ ∆𝑥 𝑢 ,𝜏𝑑𝑖𝑓𝑓≤ ∆𝑥2 𝜇 ∆𝑥≈ 10 −5

Number of Equations Stencil in RANS Solver Boundary Condition Number of equations to solve Heat Transfer 1 equation Diffusion Average solution Structural Dynamics 3 equations Fluid 5 equation mean flow + 1 ~ 7 equations for turbulence 6 ~ 12 times longer on the 1st cut assessment Each cell utilizes information from 32 adjacent cells Matrix size increase by O(N2) ~ O(N*log(N)) Requires more memory space N*log(N) operation count increase More time to swap and access memories in the cache. Shorter time step requirement. Takes loner to converge – slower propagation speed. Stronger non-linearity Convection + diffusion Diffusion: averaged quantity Convection: Dependent on the previous value The scheme becomes extremely dissipative without proper resolution of the current state Requires fine resolution to stay within acceptable accuracy Typical error analysis Error ~ O(Δ2), where 2nd order is a standard error acceptance Resolution of turbulence requires exponentially larger number of cells. Boundary Condition

Useful Concepts in CFD

Residual 𝜕𝑓(𝑥) 𝜕𝑡 +𝑢(𝑥) 𝜕𝑓(𝑥) 𝜕𝑥 −𝜇 𝜕2𝑓(𝑥) 𝜕𝑥2 =0 For a given PDE, 𝜕𝑓(𝑥) 𝜕𝑡 +𝑢(𝑥) 𝜕𝑓(𝑥) 𝜕𝑥 −𝜇 𝜕2𝑓(𝑥) 𝜕𝑥2 =0 Focus on the steady state form of the PDE, 𝑢(𝑥) 𝜕𝑓(𝑥) 𝜕𝑥 −𝜇 𝜕2𝑓(𝑥) 𝜕𝑥2 =0 Take any x0 value and plug it into the steady state part of the PDE. If the x0 is not a solution, the value in the left hand side of the equation will not be equal to zero. 𝑢(𝑥0) 𝜕𝑓(𝑥0) 𝜕𝑥 −𝜇 𝜕2𝑓(𝑥0) 𝜕𝑥2 ≠0 Then, the amount of deviation is called residual. 𝑅 𝑥0 = 𝑢(𝑥0) 𝜕𝑓(𝑥0) 𝜕𝑥 −𝜇 𝜕2𝑓(𝑥0) 𝜕𝑥2 ≠0

CFL Number For a given PDE, After applying discretization, 𝜕𝑓(𝑥) 𝜕𝑡 +𝑢(𝑥) 𝜕𝑓(𝑥) 𝜕𝑥 =0 After applying discretization, 𝑓(𝑛+1,𝑖) −𝑓(𝑛,𝑖) Δ𝑡 +𝑢(𝑥) 𝑓(𝑛,𝑖+1) −𝑓(𝑛,𝑖) Δ𝑥 =0 After rearrangement, 𝑓 𝑛+1,𝑖 =𝑓 𝑛,𝑖 −𝑢(𝑥) Δ𝑡 Δ𝑥 (𝑓 𝑛,𝑖+1 −𝑓 𝑛,𝑖 )=0 The time to spatial discretization ratio is called Courant-Friedrichs-Lewy (CFL) number. CFL number larger than 1 corrupts the obtained solution – it calculates physical impossible solution. The full numerical domain of dependence must contain the physical domain of dependence. CFL=𝑢(𝑥) Δ𝑡 Δ𝑥 <1

Relaxation 𝐴𝑥=𝑏 𝐴=𝐷+𝑈+𝐿 𝐷+ω𝐿 𝑥 𝑛+1 =ω𝑏− ω𝑈+ ω−1 𝐷 𝑥 𝑛 For a matrix form of a discretized PDE, 𝐴𝑥=𝑏 The matrix A is decomposed into a diagonal, upper triangular, and lower triangular matrices 𝐴=𝐷+𝑈+𝐿 Then, the updated value can be obtained by the following formula with a relaxation factor, ω 𝐷+ω𝐿 𝑥 𝑛+1 =ω𝑏− ω𝑈+ ω−1 𝐷 𝑥 𝑛 In simple terms, it is a measure of how much correction to apply for the current update. In other perspective, it is a measure of mixing old and the new solution . If 0<ω<1, it is called under-relaxation. If 1<ω<2, it is called over-relaxation.

Multigrid Most advanced convergence acceleration technique along with preconditioner. Achieves small scale and large scale convergence at the same time. V-cycle and W-cycle typically used

Back-Up

Physical Approximation Error 1/2 Viscous Governing Equation Reality Inviscid Governing Equation Turbulence Governing Equation

Physical Approximation Error 2/2 Forward difference y 𝑥0+Δ𝑥 =𝑦 𝑥0 + Δ𝑥 ∗ 𝑦 ′ 𝑥0 + Δ𝑥 2 2 𝑦 ′′ 𝑥0 + remainders, where x is near x0 y 𝑥0+Δ𝑥 −𝑦 𝑥0 = Δ𝑥 ∗ 𝑦 ′ 𝑥0 + Δ𝑥 2 2 𝑦 ′′ 𝑥0 𝑦 ′ 𝑥0 = 𝑦 𝑥0+Δ𝑥 −𝑦 𝑥0 Δ𝑥 − Δ𝑥 2 𝑦 ′′ 𝑥0 Slope approximation + O(Δ𝑥) Using information from one additional node 1 stencil Central difference y 𝑥0+Δ𝑥 −𝑦 𝑥0 = Δ𝑥 ∗ 𝑦 ′ 𝑥0 + Δ𝑥 2 2 𝑦 ′′ 𝑥0 + Δ𝑥 3 6 𝑦 ′′′ 𝑥0 y 𝑥0−Δ𝑥 −𝑦 𝑥0 = −Δ𝑥 ∗ 𝑦 ′ 𝑥0 + −Δ𝑥 2 2 𝑦 ′′ 𝑥0 + −Δ𝑥 3 6 𝑦 ′′′ 𝑥0 𝑦 ′ 𝑥0 = 𝑦 𝑥0+Δ𝑥 −𝑦 𝑥0−Δ𝑥 2Δ𝑥 − Δ𝑥 2 6 𝑦 ′′′ 𝑥0 Slope approximation + O(Δ𝑥2) Using information from two additional node 2 stencils More stencils Higher order scheme possible Computationally costly Boundary condition not well defined How to convert from continuous formula to discrete formula?

Finite Volume Formulation 1/2

Finite Volume Formulation 2/2

Finite Difference

Finite Element 1/2

Finite Element 2/2

Time Scale Time scale Physical length Flow of information Start-up Too small or too large Flow of information Diffusion: Average. Less sensitive to spatial resolution. Convection: Directional flow. Finer spatial resolution required to satisfy conservation. Coarse grid introduces excessive loss. Start-up Reaching the steady state may take a long time, depending on the start-up condition Numerical instability CFL number CSV (C/Sum(G)) Reduce time step Lack of damping Increase relaxation factor Slower update Artificial damping Modification of the governing equation to achieve better convergence condition, but with same order of accuracy

Condition Number Condition number in general means a measure of how much the output changes for a small change in the input. For a linear system, Ax = b Condition number for the matrix A in Ax = b gives a bound on how inaccurate the solution will be after approximation. The rate at which the solution x will change with respect to the change in b. It is defined as the maximum ratio of the relative error in x to the relative error in b. Pre-conditioner generally reduces the condition number of the matrix Faster convergence. Solution with less error.