B-char: an efficient (and feasible!) approach for mass...

B-char: an efficient (and feasible!)approach for mass-conserving characteristic

schemes in 2D and 3D

J. Droniou (Monash University)

Monash Workshop on Numerical Differential Equations 2020

Joint work with Hanz M. Cheng (formerly Monash, now EindhovenUniversity of Technology)

1 The problem: numerical methods with inexact calculations

2 B-char method: cheap, and perfectly mass conservative

3 Numerical tests2D tests3D tests

Linear advection model

φ∂c

∂t+ div(uc) = 0 on QT := Ω× (0,T ),

c(·, 0) = cini on Ω.

Ω: polygonal/polyhedral domain, with mesh M.

φ: porosity, 0 < φ∗ ≤ φ ≤ φ∗, piecewise constant on mesh.

u: Darcy velocity, u ∈ L∞(0,T ; L2(Ω)), divu = 0 andu · n = 0 on ∂Ω.

cini: initial concentration, cini ∈ L∞(Ω).

ELLAM method

Time steps: Time discretisation

0 = t(0) < t(1) < . . . < t(N) = T , with δt(n+ 12

) = t(n+1) − t(n).

Let u(n+1) ∈ L2(Ω)d approximate u on (t(n), t(n+1)), withdivu(n+1) = 0 and u(n+1) · n = 0 on ∂Ω.

Test function: ψ satisfying

φ∂ψ

∂t+u(n+1) ·∇ψ = 0 on Ω×(t(n), t(n+1)) , ψ(·, t(n+1)) given.

I Set Ft(x) flow of u(n+1)/φ, that is

dFt(x)

u(n+1)(Ft(x))

φ(Ft(x)), F0(x) = x .

Thenψ(x , t(n)) = ψ(F

δt(n+ 12 )(x), t(n+1)).

Time stepping in ELLAM (=Eulerian Lagrangian LocalisedAdjoint Method):

Ωφ(x)(cψ)(x , t(n+1)) dx =

Ωφ(x)(cψ)(x , t(n)) dx

ELLAM method

0 = t(0) < t(1) < . . . < t(N) = T , with δt(n+ 12

) = t(n+1) − t(n).

φ∂ψ

dFt(x)

u(n+1)(Ft(x))

φ(Ft(x)), F0(x) = x .

δt(n+ 12 )(x), t(n+1)).

Ωφ(x)(cψ)(x , t(n+1)) dx =

ELLAM method

0 = t(0) < t(1) < . . . < t(N) = T , with δt(n+ 12

) = t(n+1) − t(n).

φ∂ψ

dFt(x)

u(n+1)(Ft(x))

φ(Ft(x)), F0(x) = x .

δt(n+ 12 )(x), t(n+1)).

Ωφ(x)(cψ)(x , t(n+1)) dx =

ELLAM method

φ∂ψ

dFt(x)

u(n+1)(Ft(x))

φ(Ft(x)), F0(x) = x .

δt(n+ 12 )(x), t(n+1)).

Ωφ(x)(cψ)(x , t(n+1)) dx =

ELLAM method: global and local mass conservation

Global mass conservation: make ψ(x , t(n+1)) ≡ 1:

Ωφ(x)c(x , t(n+1)) dx =

Ωφ(x)c(x , t(n)) dx .

Local mass conservation: since divu = 0,

If c(·, t(n)) = 1 then c(·, t(n+1)) = 1.

ELLAM for piecewise constant approximations

I At each time, we are looking for ch(·, t(n)) = (c(n)M )M∈M

piecewise constant approximation of c on M.

I Notation: the porous volume in a set A is

|A|φ =

ELLAM formulation: take ψ(·, t(n+1)) = 1K for a cell K ∈M:

|K |φc(n+1)K =

M∈M|M ∩ F

−δt(n+ 12 )(K )|φc(n)

Global and local mass conservation

|K |φc(n+1)K =

M∈M|M ∩ F

−δt(n+ 12 )(K )|φc(n)

Global mass conservation: OK by summing over K and using

K∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |M|φ.

Local mass conservation: OK because

M∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |F

−δt(n+ 12 )(K )|φ = |K |φ.

|K |φc(n+1)K =

M∈M|M ∩ F

−δt(n+ 12 )(K )|φc(n)

K∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |M|φ.

M∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |F

−δt(n+ 12 )(K )|φ = |K |φ.

|K |φc(n+1)K =

M∈M|M ∩ F

−δt(n+ 12 )(K )|φc(n)

K∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |M|φ.

M∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |F

−δt(n+ 12 )(K )|φ = |K |φ.

ELLAM in practice: what needs to be computed

Transport of cells: K polygonal/polyhedral cell, but F−δt(n+ 1

2 )(K )

is a generic potato, that needs to be approximated...

F−δt(n+

12)(K)

t(n+1)

Figure: Exact (left) and approximated (right) trace-back of K .

ELLAM in practice: what needs to be computed

Intersection of regions: need to compute (porous volume of)M ∩ F

−δt(n+ 12 )(K ).

I Algorithms for areas of intersections of polygons (2D) are ok,but expensive.

I Algorithms for volume of intersections of polyhedras (3D) areterrible!

ELLAM in practice: revisiting mass conservation

I Global and local mass conservation are based on∑

K∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |M|φ (global),

M∈M|M ∩ F

−δt(n+ 12 )(K )|φ = |F

−δt(n+ 12 )(K )|φ = |K |φ (local).

I Issue: we only compute K , and

|M ∩ K |φ ≈ |M ∩ F−δt(n+ 1

2 )(K )|φ.

Not a problem for global mass conservation (as (K )K∈M forms apartition of the domain), but breaks down local massconservation...

An original idea...

Approximate polygons/polyhedras by balls,

track balls(keeping them as balls), intersect balls.

An original idea...

Approximate polygons/polyhedras by balls, track balls(keeping them as balls),

intersect balls.

An original idea...

Approximate polygons/polyhedras by balls, track balls(keeping them as balls), intersect balls.

... that needs to be enhanced!

I Loss of volume in K when approximating by balls (gaps), andloss of volume when intersecting balls.

I Very inaccurate approximation of K (and thus of F−δt(n+ 1

2 )(K ))

by tracked balls.

bad solutions, clearly not conserving mass.

Initial adjustments

I Cell K with balls (BK ,s)s=1,...,nK .

Distribution of porous volume: introduce porous density ρK ,constant during evolution, such that

|BK ,s |φ = |K |φ.

I ρK |BK ,s |φ equivalent porous volume inside ball.

Tracking of balls: assuming φ constant, the volume (and radius)of BK ,s remains constant during tracking (generalised Liouvilletheorem).

Initial adjustments

|BK ,s |φ = |K |φ.

Initial adjustments

|BK ,s |φ = |K |φ.

Initial adjustments

Intersections of balls without loss of mass: straight intersectionof balls in K and M leads to

|K ∩M|φ ≈∑

ρMφM |BK ,s ∩ BM,m|.

I But loss of mass through intersection of balls. So we computethe fraction of mass of BK ,s that comes from BM,m:

fK ,s,M,m =ρMφM |BK ,s ∩ BM,m|∑

L∈M∑nL

`=1 ρLφL|BK ,s ∩ BL,`|

and we set

|M ∩ K |φ ≈ VK ,M

ρK φK ,s |BK ,s |nM∑

fK ,s,M,m.

Initial adjustments

Intersections of balls without loss of mass: straight intersectionof balls in K and M leads to

|K ∩M|φ ≈∑

ρMφM |BK ,s ∩ BM,m|.

I But loss of mass through intersection of balls. So we computethe fraction of mass of BK ,s that comes from BM,m:

fK ,s,M,m =ρMφM |BK ,s ∩ BM,m|∑

L∈M∑nL

`=1 ρLφL|BK ,s ∩ BL,`|

and we set

|M ∩ K |φ ≈ VK ,M

ρK φK ,s |BK ,s |nM∑

fK ,s,M,m.

Mass conservations?

Local mass conservation: came from∑

|M ∩ F−δt(n+ 1

2 )(K )|φ = |F−δt(n+ 1

2 )(K )|φ = |K |φ.

We therefore need∑

= |K |φ. OK because∑

∑m fK ,s,M,m = 1.

Global mass conservation: came from∑

|M ∩ F−δt(n+ 1

2 )(K )|φ = |M|φ.

= |M|φ. KO!

Mass conservations?

|M ∩ F−δt(n+ 1

2 )(K )|φ = |F−δt(n+ 1

2 )(K )|φ = |K |φ.

∑m fK ,s,M,m = 1.

|M ∩ F−δt(n+ 1

2 )(K )|φ = |M|φ.

= |M|φ. KO!

Mass conservations?

|M ∩ F−δt(n+ 1

2 )(K )|φ = |F−δt(n+ 1

2 )(K )|φ = |K |φ.

∑m fK ,s,M,m = 1.

|M ∩ F−δt(n+ 1

2 )(K )|φ = |M|φ.

= |M|φ. KO!

Mass conservations?

|M ∩ F−δt(n+ 1

2 )(K )|φ = |F−δt(n+ 1

2 )(K )|φ = |K |φ.

∑m fK ,s,M,m = 1.

|M ∩ F−δt(n+ 1

2 )(K )|φ = |M|φ.

= |M|φ. KO!

Second adjustment: redistributions

Global:∑

= |M|φ. Local:∑

= |K |φ.

I Step 2: redistribute to get local mass conservation

V(n+1)

|K |φ∑

L V(n+ 1

V(n+ 1

Global:∑

= |M|φ. Local:∑

= |K |φ.

I Step 0: set V(0)

K ,M= V

V(n+1)

|K |φ∑

L V(n+ 1

V(n+ 1

Global:∑

= |M|φ. Local:∑

= |K |φ.

For n = 0, . . . ,N, iterate:

I Step 1: redistribute to get global mass conservation

V(n+ 1

|M|φ∑R V

V(n+1)

|K |φ∑

L V(n+ 1

V(n+ 1

Global:∑

= |M|φ. Local:∑

= |K |φ.

For n = 0, . . . ,N, iterate:

I Step 1: redistribute to get global mass conservation

V(n+ 1

|M|φ∑R V

V(n+1)

|K |φ∑

L V(n+ 1

V(n+ 1

I Error in global/local mass tends to reduce at each iteration...but very slowly after the first few steps.

Achieving exact conservation: after n ∼ 10

I Error in global/local mass tends to reduce at each iteration...but very slowly after the first few steps.

Achieving exact conservation: after n ∼ 10, stop iterations and

find, in the vicinity of the current (V(n)

K ,M)K ,M , one solution to the

global and local mass conservation equations.

Achieving exact conservation: after n ∼ 10:

Find x = (xK ,M

)K ,M such that:

((1 + xK ,M

K ,M)K ,M exactly satisfies the global and

local mass balance equations,

0 ≤ 1 + xK ,M≤ 2,

|x|2 is minimal.

Then, use VK ,M

= (1 +xK ,M

K ,Mas porous volumes of cell

intersections.

I (xK ,M

)K ,M are ]cells× ]cells unknowns, but the actual

minimisation problem is much smaller (only a few V(n)

K ,Mare

non-zero).

Achieving exact conservation: after n ∼ 10:

Find x = (xK ,M

)K ,M such that:

((1 + xK ,M

K ,M)K ,M exactly satisfies the global and

local mass balance equations,

0 ≤ 1 + xK ,M≤ 2,

|x|2 is minimal.

Then, use VK ,M

= (1 +xK ,M

K ,Mas porous volumes of cell

intersections.

I (xK ,M

)K ,M are ]cells× ]cells unknowns, but the actual

minimisation problem is much smaller (only a few V(n)

K ,Mare

non-zero).

Comparison with “polygonal” ELLAM: translation

I “Polygonal” ELLAM: classical approach, computing K andintersection M ∩ K .I B-char: 4 balls in each cell.

Test case: Ω = (0, 1)2, cini = 1 on ( 116 ,

516 )× ( 1

16 ), velocityu = ( 1

16 , 0), final time T = 8.

516 )× ( 1

16 , 0), final time T = 8.

Figure: 16×16 grid, δt = 0.8 (left: polygonal; right: B-char).

516 )× ( 1

16 , 0), final time T = 8.

516 )× ( 1

16 , 0), final time T = 8.

516 )× ( 1

16 , 0), final time T = 8.

Polygonal B-char

Mesh δt CPU (1 step) L2 error CPU (1 step) L2 error

16× 16 0.8 0.5s 3.7e-01 0.1s 3.8e-01

32× 32 0.4 6.5s 3.2e-01 0.4s 3.3e-01

64× 64 0.2 97.4s 2.7e-01 3.5s 2.9e-01

Table: CPU runtime and errors

Comparison with “polygonal” ELLAM: rotation

Test case: Ω = (0, 1)2, cini = 1 on disc of center ( 14 ,

34 ) and

radius 18 , final time T = 8. Streamlines of velocity:

Figure: Streamlines of the velocity fieldu = ((1− 2y)(x − x2),−(1− 2x)(y − y2)).

Test case: Ω = (0, 1)2, cini = 1 on disc of center ( 14 ,

34 ) and

radius 18 , final time T = 8.

Figure: Initial condition (left), final solution (right).

Results:

Polygonal B-char

Mesh δt CPU (1 step) L2 error CPU (1 step) L2 error

16× 16 0.8 2.7s 5.1e-01 0.2s 5.1e-01

32× 32 0.4 43s 4.2e-01 1.3s 4.1e-01

64× 64 0.2 701s 3.6e-01 14.5s 3.6e-01

Table: CPU runtime and errors

Solid body rotation

Velocity: simple rotation around the center of Ω = (0, 1)2.

Figure: Solid body rotation on a 128×128 mesh (left: initial condition;right: numerical solution at T = 2π).

I Underlying ELLAM discretisation allows for larger time stepsδt = 2π

10 (in literature, usually, δt ≤ 2π810 ).

Deformational flow

Velocity: velocity reverses at half-time T/2:

u = (sin2(πx) sin(2πy) cos(πt/T ),− sin2(πy) sin(2πx) cos(πt/T )).

Deformational flow

Results:

Figure: 64× 64 mesh, δt = 0.5 (left: initial condition; right: numericalsolution at T = 5).

Deformational flow

Results:

Figure: 128× 128 mesh, δt = 0.25 (left: initial condition; right:numerical solution at T = 5).

Deformational flow

Results:

Figure: At halftime T = 2.5 (left: 64× 64 cells; right: 128× 128 cells).

Setting

I Ω = (0, 1)3, T = 8.

I B-char with 8 balls per cell, 163 mesh, δt = 0.8.

I 3 test cases:

1. Piecewise constant cini in cube, velocity: translation in x .

2. Piecewise constant cini in cylinder, velocity: rotation &stretching in (x , y), translation in z .

3. Continuous bump cinit, same velocity as in 2.

Results

Test case δt CPU time L1 error L2 error(one time step)

1 0.8 37.2s 4.8e-01 4.1e-01

2 0.8 63.5s 9.6e-01 6.2e-01

3 0.8 63.2s 2.4e-01 2.4e-01

Table: CPU runtime and errors in 3D.

Bibliography

Main paper:H. M. Cheng and J. Droniou, “An efficient implementation of mass conservingcharacteristic-based schemes in 2D and 3D”. To appear in SIAM J. Sci. Comput.

Other references:M. A. Celia, T. F. Russell, I. Herrera, and R. E. Ewing. “An Eulerian-Lagrangianlocalized adjoint method for the advection-diffusion equation”. Adv. Water Resources,13(4):187–206, 1990.

T. Arbogast and C. Huang. “A fully mass and volume conserving implementation of acharacteristic method for transport problems”. SIAM J. on Sci. Comput.,28(6):2001–2022, 2006.

T. Arbogast and C.-S. Huang. “A fully conservative Eulerian-Lagrangian method for aconvection-diffusion problem in a solenoidal field”. J. Comput. Phys.,229(9):3415–3427, 2010.

H. M. Cheng, J. Droniou, and K.-N. Le. “Convergence analysis of a family of ELLAMschemes for a fully coupled model of miscible displacement in porous media”. Numer.Math., 141(2):353–397, 2019.

H. M. Cheng, J. Droniou, and K.-N. Le. “A combined GDM–ELLAM–MMOC schemefor advection dominated PDEs”. https://arxiv.org/abs/1805.05585, 35p, 2018.

M. D’Elia, D. Ridzal, K. J. Peterson, P. Bochev, and M. Shashkov.“Optimization-based mesh correction with volume and convexity constraints”. J.Comput. Phys., 313:455–477, 2016.

Thanks.

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