Comparison of Nonlinear and Linear StabilizationSchemes for Advection-Diffusion Equations

Ryan Grove*, Timo HeisterDepartment of Mathematical Sciences

Clemson University*rgrove@clemson.edu

Abstract

Standard finite element discretizations of advection-diffusion equations introduceunphysical oscillations around steep gradients. Therefore, stabilization must beadded to the discrete formulation to obtain correct solutions. The SUPG, dCG91,and Entropy Viscosity schemes are compared using stationary and non-stationarytest equations. Differences in maximum overshoot and undershoot, smear, andconvergence orders are compared using code written using deal.ii.

Motivation

Here we consider a composition field, which describes the evolution ofadditional fields that are transported along with the velocity and may reactwith each other and react to other features of the solution, but that do notdiffuse. Below we see an example in the form of a rising bubble.

Start End

Objective:The rising bubble should look exactly the same when it is done beingdisplaced as it does at the very beginning.

Things that go wrong:• Diffusion - Should be a perfect rectangle• Overshoots - The maximum value should be no higher than 1• Undershoots - The minimum value should be no lower than 0

Note that it is difficult to minimize overshoots and undershoots withoutcreating too much smear.

Model

We assume that Ω is a bounded domain in Rd, d = 2, 3, with Lipschitzboundary, and consider the parabolic initial-boundary value problem: for eacht ∈ [0, T ] find u(x, t) such that

∂u(x, t)

∂t−∇ · (ε∇u(x, t)) +

∇ · (βu(x, t)) = f in QT := Ω× [0, T ] (1)

u(x, t) = 0 in ΣT := ∂Ω× [0, T ] (2)

u(x, 0) = u0(x) on Ω (3)

where β(x, t) : Ω× [0, T ]→ Rdim is a divergence free convection direction,ε > 0 is a diffusion coefficient, and solution u(x, t) : Ω× [0, T ]→ R.

Time Discretization

Consider Backwards Euler:uk − uk−1

∆tk= F (x, tk, uk), (4)

where ∆tk = tk − tk−1 = 1c. Therefore, from (1), we have that

cuk −∇ · (ε∇uk) +∇ · (βuk) = f + cuk−1 = f ′. (5)

We then proceed using the Finite Element Method with piecewise linear elements,and we obtain our weak form which is:

c (φ, uk) + (∇φ, ε∇uk)− (∇φ, βuk) =(φ, f′

), (6)

Note that the BDF2 method was also used and that it creates a differently definedf′ and c, but still has to solve a problem of this form.

First Order Viscosity

This is a naive method that simply adds more diffusion to the problem wherethere is advection. Adding

(∇φ, εh∇uk)to our weak form yields:

c (φ, uk) + (∇φ, (ε+ εh)∇uk)− (∇φ, βuk) = (φ, f’) , (7)

where εh = chhK||β||L∞(K).

SUPG[1],[2] (Streamline Upwinding Petrov Galerkin)

This method attempts to add stabilization in the directions of streamlines of theadvection field. Added to our problem is the term(

τβ · ∇φ,−f + β · ∇uk − ε∆uk +uk − uk−1

∆tk

),

where τ is actually the following:

α(hK/k)

2||β||L∞(K)

(coth

(||β||L∞(K)(hK/k)

2ε

)−

2ε

||β||L∞(K)(hK/k)

)and is computed on each cell, where α is a user-picked O(1) constant and k isthe finite element order.

Entropy Viscosity[3]

This method aims to reduce overshoots through adding first order viscosity inplaces of high entropy using a nonlinear scheme (we use fixed point iteration inthe stationary case, and the solution of the previous time step in the nonstationarycase).Added to our problem is the term

(∇φ, vh(uk)∇uk) ,where

vh(uk) = max (vmax(uk), vE(uk)) .

to the equation yielding:

c (φ, uk) + (∇φ, (ε+ vh(uk))∇uk)− (∇φ, βuk) = (φ, f’) .

The above has:vmax(uk) = αmaxhK||β||L∞(K),

which is essentially First Order Viscosity, and

vE(uk) = αEh2K||rE(uk)||L∞(K)

||E(uk)− Eavg||L∞(Ω)

where αE is a user-picked O(1) constant, ||E(uk)−Eavg||L∞(Ω) is the globalentropy variation, E(u) = (u− uavg)

2, and ||rE(uk)||L∞(K) is as follows:

||∂E(uk)

∂t+ (uk − uavg) (β · ∇uk − ε∆uk − f) ||L∞(K)

dCG91[1],[2]

This method aims to reduce overshoots through a nonlinear scheme (we use fixedpoint iteration in the stationary case, and the solution of the previous time stepin the nonstationary case). In addition to SUPG, added to our problem is theterm

(∇φ, ε(uk)∇uk)where

ε(u) = α′max

0,τ ||β||L∞(K)||r(u)||L∞(K)

|∇u|L∞(K)

− τ||r(u)||2L∞(K)

|∇u|2L∞(K)

,

where α′ is a user-picked O(1) constant and

r(u)L∞(K) = ||β · ∇uk − ε∆uk +uk − uk−1

∆tk− f||L∞(K).

Exact Solution (Time Independent)

β =(cos(−π

3), sin(−π

3))

ε = 10−5

Ω = [0, 1]2

f = 0u = 0, if y ≤ −

√3x+ .7 or u = 1, otherwise

Smear and Max Over/Undershoot of Linear Elements

Note: All calculations were performed on a personal Linux laptop. Code waswritten using deal.ii[4] library. The following are results for when the numberof global refinements, n, is 6, and ε = 1e− 5:

Method Parameter Value Max val Max OU Smear L2 ErrorNo Stabilization - 1.309 3.1516e-1 3.7000e-2 7.2364e-2

FOV 1.0 1.0 0.0 3.5800e-1 1.3885e-1SUPG 1.0 1.086 6.1433e-2 4.4000e-2 5.3867e-2

dCG91 Nonlinear 0.01 1.064 2.4761e-2 5.7000e-2 5.6957e-2EV Nonlinear 1.0(max), 0.1(E) 1.145 7.0485e-2 4.3000e-2 5.1836e-2

Summary of Results for Linear Elements

From top left to bottom right: Exact solution, No stabilization, FOV, SUPG,dCG91, and EV

Exact Solution (Time Dependent)

β = (−2πy, 2πx)ε = 10−5

Ω = [0, 1]2

f = 0u = 0, if y ≤ −

√3x+ .7 or u = 1, otherwise

Smear and Max Over/Undershoot of Linear Elements

Note: All calculations were performed on a personal Linux laptop. Code waswritten using deal.ii[4] library. The following are the results for one rotation,where the number of time steps is 1000, n = 6, and ε = 1e− 5:

Method Parameter Value Max OU Smear L2No Stability - 4.899011e-1 2.990000e-1 1.999698e-1

FOV 1.0 0.0 2.550000e-1 4.833988e-1SUPG 1.0 5.086724e-2 3.630000e-1 1.894290e-1dCG91 0.01 3.387065e-4 3.740000e-1 2.249393e-1

EV 1.0(max), 0.1(E) 9.174072e-2 3.710000e-1 2.074987e-1

Summary of Results for Linear Elements

From top left to bottom right: Exact Solution, No stabilization, FOV, SUPG,dCG91, and EV

Bibliography

[1] John, Volker, and Petr Knobloch. ”On spurious oscillations at layersdiminishing (SOLD) methods for convection-diffusion equations: Part I-Areview.” Computer Methods in Applied Mechanics and Engineering 196.17(2007): 2197-2215.

[2] John, Volker, and Petr Knobloch. ”On spurious oscillations at layersdiminishing (SOLD) methods for convection-diffusion equations: PartII-Analysis for P1 and Q1 finite elements.” Computer Methods in AppliedMechanics and Engineering 197.21 (2008): 1997-2014.

[3] Guermond, Jean-Luc, Richard Pasquetti, and Bojan Popov. ”Entropyviscosity method for nonlinear conservation laws.” Journal of ComputationalPhysics 230.11 (2011): 4248-4267.

[4] Wolfgang Bangerth, Timo Heister, et al. The deal.II Library, Version 8.2.Archive of Numerical Software, vol. 3, 2015, DOI10.11588/ans.2015.100.18031, 2015. https://www.dealii.org/

This work is partially supported by the Computational Infrastructure in Geodynamics initiative (CIG), through the National Science Foundation under Award No. EAR-0949446 and The University of California – Davis.