Convergence in Computational Science 10/28/2018 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: turbulence models Geometry modeling error: is the geometry good enough to capture the actual physics? Round and corners Is far-field boundary far enough? 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 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 Lower level programming error System configuration error 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. It satisfies the applied discretization 1st order, 2nd order Upwind, downwind Does it follow physical flow of information? Shock capturing has become successful only after taking this effect into consideration Numerical damping Limiter Additional terms to satisfy convergence criteria Precision How much precision to trust? 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 Expansion of a differentiable function 1st order form 2nd order formulation Accuracy of y(x) depends on the relative size of (x-x0), y’(x), and y’’(x0) Where y’(x0) or y’’(x0) is non-negligible, reduce (x-x0) Where y’(x0) or y’’(x0) is near zero, moderate size of (x-x0) is acceptable How large is acceptable? y x =𝑦 𝑥0 + 𝑥−𝑥0 ∗ 𝑦 ′ 𝑥0 + 𝑥−𝑥0 2 𝑦 ′′ 𝑥0 + remainders, where x is near x0 y x =𝑦 𝑥0 + 𝑥−𝑥0 ∗ 𝑦 ′ 𝑥0 , where x is near x0 y x =𝑦 𝑥0 + 𝑥−𝑥0 ∗ 𝑦 ′ 𝑥0 + 𝑥−𝑥0 2 𝑦 ′′ 𝑥0 , where x is near 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. Finer grid does not always guarantee convergence, due to the mixed length scale. 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 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) Requires more memory space N*log(N) operation count increase More time to swap and access memories in the cache. Takes loner to converge 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

Back-Up

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