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Qualitatively reliable numerical models of time-dependent problems Istv´ an Farag´ o Farkas Mikl´ os Seminar September 21, 2017, Budapest, Hungary Farag´oIstv´ an (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 1 / 54

Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

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Page 1: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Qualitatively reliable numerical models oftime-dependent problems

Istvan Farago

Farkas Miklos Seminar

September 21, 2017,

Budapest, Hungary

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 1 / 54

Page 2: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Joint work with ......

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 2 / 54

Page 3: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Outline

1 Introduction and motivation

2 Linear parabolic problemsPartial differential operators of parabolic typeQualitative properties in the discrete model

3 Two-levels discrete models

4 Heat conduction problemHeat equation in 1D

5 The Crank-Nicolson discretized heat conduction modelCN-method in 1DCN-method in IRd

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 3 / 54

Page 4: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Outline

1 Introduction and motivation

2 Linear parabolic problemsPartial differential operators of parabolic typeQualitative properties in the discrete model

3 Two-levels discrete models

4 Heat conduction problemHeat equation in 1D

5 The Crank-Nicolson discretized heat conduction modelCN-method in 1DCN-method in IRd

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 3 / 54

Page 5: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Outline

1 Introduction and motivation

2 Linear parabolic problemsPartial differential operators of parabolic typeQualitative properties in the discrete model

3 Two-levels discrete models

4 Heat conduction problemHeat equation in 1D

5 The Crank-Nicolson discretized heat conduction modelCN-method in 1DCN-method in IRd

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 3 / 54

Page 6: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Outline

1 Introduction and motivation

2 Linear parabolic problemsPartial differential operators of parabolic typeQualitative properties in the discrete model

3 Two-levels discrete models

4 Heat conduction problemHeat equation in 1D

5 The Crank-Nicolson discretized heat conduction modelCN-method in 1DCN-method in IRd

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 3 / 54

Page 7: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Outline

1 Introduction and motivation

2 Linear parabolic problemsPartial differential operators of parabolic typeQualitative properties in the discrete model

3 Two-levels discrete models

4 Heat conduction problemHeat equation in 1D

5 The Crank-Nicolson discretized heat conduction modelCN-method in 1DCN-method in IRd

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 3 / 54

Page 8: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

1. Introduction and motivation

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 4 / 54

Page 9: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Modelling process

Physical model → Mathematical model → Numerical model

The mathematical model and numerical model on the fixed mesh shouldpreserve the basic (physically motivated) qualitative properties

Convergence for the numerical model is a necessary requirement. BUT:Not only the condition of the convergence should be taken into theconsideration.

What is the relation in the numerical model between the conditions of theconvergence and the qualitative preservation?

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 5 / 54

Page 10: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Numerical solution and qualitative properties

What are we expecting from the (numerical) solution?

to possess some adequate qualitative properties,

to be a good approximation of the real solution ⇒ convergence

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 6 / 54

Page 11: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Physical properties of heat conduction processes

Heat conduction phenomenon (without source) in 1D with homogeneousfirst boundary condition. Adequate behaviour of the process

Figure: Heat conduction process

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 7 / 54

Page 12: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Figure: Nonnegativity

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 8 / 54

Page 13: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Figure: Increase of the maximum

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 9 / 54

Page 14: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Different qualitative properties!!! The most characteristic (physicallybased) properties:

non-negativity preservation

maximum-minimum principle

contractivity in maximum norm

There are many others, e.g.

shape preservation

monotone decrease in time

sign-stability, etc.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 10 / 54

Page 15: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Numerical Example

Problem: two-dimensional heat equation on the unit square withhomogeneous boundary condition.

∂v

∂t−

2∑l=1

∂2v

∂x2l

= 0, g = 0

Ω = [0, 1]× [0, 1],

Numerical method:FEM,bilinear elements

step-sizes in space:∆x = 1/10, ∆y = 1/12,

θ = 1/2, Crank-Nicolson(absolute stable!)

initial function:

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 11 / 54

Page 16: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Numerical Example – Results

Approximations at the 10th time level.

∆t = 0.1

∆t = 0.0005

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 12 / 54

Page 17: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Numerical Example – Results

Approximations at the 10th time level.

∆t = 0.1 ∆t = 0.0005

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 12 / 54

Page 18: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Numerical Example – Results (cont’d)

Approximation at the 10th timelevel.

∆t = 0.0001

The nonnegativity of the initialtemperature is not preserved.

Other problems:

1 for ∆t = 0.0001 it breaksthe maximum-minimumprinciple and the maximumnorm contractivity property

2 for ∆t = 0.1 it producesspurious oscillations

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 13 / 54

Page 19: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Numerical Example – Results (cont’d)

Approximation at the 10th timelevel.

∆t = 0.0001

The nonnegativity of the initialtemperature is not preserved.

Other problems:

1 for ∆t = 0.0001 it breaksthe maximum-minimumprinciple and the maximumnorm contractivity property

2 for ∆t = 0.1 it producesspurious oscillations

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 13 / 54

Page 20: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

2. Linear parabolic problems

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 14 / 54

Page 21: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Partial differential operators of parabolic type

Definition of the solution domain for the parabolic problems:

Figure: Solution domain

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 15 / 54

Page 22: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

We consider the linear partial differential operator

L ≡ ∂

∂t−

∑0≤|ς|≤δ

aς∂|ς|

∂ς1x1 . . . ∂ςd xd≡ ∂

∂t−

∑0≤|ς|≤δ

aςDς , (1)

For the constant function: L(const) = a0 · const

Special case: diffusion operator of the form

Ldiff (v) ≡ ∂v

∂t−

d∑l=1

∂2v

∂x2l

(a0 = 0)

Hence: Ldiff (const) = 0.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 16 / 54

Page 23: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Maximum-minimum principles

Definition

Weak maximum-minimum principle (WMP): for any function v ∈ dom L

min0,minΓt1v+ t1 ·min0, infQt1

Lv ≤ v(x, t1) ≤max0,maxΓt1

v+ t1 ·max0, supQt1Lv (2)

holds for all x ∈ Ω, t1 ∈ (0,T ).

Definition

Strong maximum-minimum principle (SMP): for any function v ∈ dom L

minΓt1

v + t1 ·min0, infQt1

Lv ≤ v(x, t1) ≤ maxΓt1

v + t1 ·max0, supQt1

Lv (3)

is satisfied for all x ∈ Ω, t1 ∈ (0,T ).

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 17 / 54

Page 24: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Boundary maximum-minimum principles

Definition

Weak boundary maximum-minimum principle (WBMP): for any functionv ∈ dom L and t1 ∈ (0,T ) such that Lv |Qt1

= 0 the inequalities

min0,minΓt1

v ≤ minQt1

v , max0,maxΓt1

v ≥ maxQt1

v

hold.

Definition

Strong boundary maximum-minimum principle (SBMP): for any functionv ∈ dom L and t1 ∈ (0,T ) such that Lv |Qt1

= 0 the inequalities

minΓt1

v = minQt1

v , maxΓt1

v = maxQt1

v

hold.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 18 / 54

Page 25: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Relations between the maximum-minimum principles

Theorem

Between the maximum-minimum principles the following implications arevalid.

SMP =⇒ WMP

⇓ ⇓

SBMP =⇒ WBMP

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 19 / 54

Page 26: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Definitions

Definition

Non-negativity preservation (NP): for any v ∈ dom L and t1 ∈ (0,T ) suchthat minΓt1

v ≥ 0 and Lv |Qt1≥ 0, the relation v |Qt1

≥ 0 holds.

Definition

Contractivity in maximum norm (MNC): for all arbitrary two functions v?,v?? ∈ dom L and t1 ∈ (0,T ) such that Lv?|Qt1

= Lv??|Qt1and

v?|∂Ω×[0,t1] = v??|∂Ω×[0,t1], the relation

maxx∈Ω|v?(x, t1)− v??(x, t1)| ≤ max

x∈Ω|v?(x, 0)− v??(x, 0)|

is valid.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 20 / 54

Page 27: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Basic relations

Figure: Implications for the qualitative properties

For the heat equation without source function L1 = a0 = 0 therefore allimplications are valid and the NP-property implies all the other qualitativeproperties.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 21 / 54

Page 28: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Meshes and discrete operator

Mesh function: ν: real-valued function defined on QtM.

Mesh operator: discrete linear operator L which maps as follows:

Figure: Discrete operator

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 22 / 54

Page 29: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Maximum principle

Each definition is the discrete analog of the continuous case!

Definition

Discrete weak maximum-minimum principle (DWMP):

min0,minGt1ν+ t1 ·min0, infQt1

Lν ≤ ν(xi , t1) ≤max0,maxGt1

ν+ t1 ·max0, supQt1Lν (4)

for any mesh function ν ∈ domL is satisfied for all xi ∈ P, t1 ∈ RtM.

Definition

Discrete strong maximum-minimum principle (DSMP):

minGt1

ν + t1 ·min0, infQt1

Lν ≤ ν(xi , t1) ≤ maxGt1

ν + t1 ·max0, supQt1

Lν (5)

for any mesh function ν ∈ domL is satisfied for all xi ∈ P, t1 ∈ RtM.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 23 / 54

Page 30: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Maximum principle(cont’d)

Definition

Discrete weak boundary maximum-minimum principle (DWBMP): for anyfunction ν ∈ domL and t1 ∈ RtM

such that Lν|Qt1= 0 we have

min0,minGt1

ν ≤ minQt1

ν, max0,maxGt1

ν ≥ maxQt1

ν.

Definition

Discrete strong boundary maximum-minimum principle (DSBMP): for anyfunction ν ∈ domL and t1 ∈ RtM

such that Lν|Qt1= 0 we have

minGt1

ν = minQt1

ν, maxGt1

ν = maxQt1

ν.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 24 / 54

Page 31: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Relation between the maximum principles

Theorem

Between the discrete maximum-minimum principles the followingimplications are valid.

DSMP =⇒ DWMP

⇓ ⇓

DSBMP =⇒ DWBMP

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 25 / 54

Page 32: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Non-negativity preservation and contractivity

Definition

Discrete non-negativity preservation (DNP): for any ν ∈ domL andt1 ∈ RtM

such that minGt1ν ≥ 0 and Lν|Qt1

≥ 0, the relation ν|Qt1≥ 0

holds.

Definition

Discrete contractivity in maximum norm (DMNC): for any two functionsν?, ν?? ∈ domL and t1 ∈ RtM

such that Lν?|Qt1= Lν??|Qt1

andν?|P∂×R0

t1

= ν??|P∂×R0t1

, the relation

maxxi∈P|ν?(xi , t1)− ν??(xi , t1)| ≤ max

xi∈P|ν?(xi , 0)− ν??(xi , 0)|

holds.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 26 / 54

Page 33: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Relation between the discrete qualitative properties

Connection between the discrete qualitative proerties:

Figure: Implications for the discrete operators

Here 11(xi , tn) = 1 and tt(xi , tn) = n∆t.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 27 / 54

Page 34: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Relation between the discrete qualitative properties

Therefore:If L(11) = 0 and Ltt ≥ 11 then all implications are true. Hence, underthese conditions DNP implies all the other discrete qualitative properties.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 28 / 54

Page 35: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

3. Two-levels discrete model

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 29 / 54

Page 36: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

An important example

Special discrete linear operator L:

(Lν)ni = (X1ν

n − X2νn−1)i , i = 1, . . . ,N, n = 1, . . . ,M, (6)

What are the above conditions?

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 30 / 54

Page 37: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Condition for the qualitative properties

(The operator: (Lν)ni = (X1ν

n − X2νn−1)i .)

First condition: L(11) = 0 ⇔ (X1 − X2)e = 0 (where ei = 1 for all i)

Second condition: L(tt) ≥ 11 ⇔ X1(n∆t · e)− X2((n − 1)∆t · e) ≥ e

Assume that the first condition is satisfied. If one of the conditions

X1e ≥ 1

∆te or X2e ≥ 1

∆te

is satisfied then the second condition is also valid.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 31 / 54

Page 38: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

4. Heat conduction problem

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 32 / 54

Page 39: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Continuous and discrete equation in 1D

Let us consider the 1D parabolic initial boundary value problem

Lv ≡ ∂v

∂t− ∂2v

∂x2= f , in (0, 1)× (0,T ) (7)

v(0, t) = v(1, t) = 0, (8)

v(x , 0) is given . (9)

This problem preserves the nonnegativity, fulfills the strongmaximum-minimum principle and contractive in maximum norm.

The numerical solution:

Mνn+1 − νn

∆t+ θKνn+1 + (1− θ)Kνn = η(n,θ), (10)

where η(n,θ) := θηn+1 + (1− θ)ηn, and θ is a parameter from the interval[0, 1].

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 33 / 54

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Conditions for the qualitative properties

X1 − X2 =1

hK

and

(X1 − X2)e = 0. X2e =1

h∆tMe =

1

∆te.

Both conditions are satisfied!

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 34 / 54

Page 41: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Relations between the basic qualitative properties:

DSMP ⇔ DNP

DMNC

If the DNP property is satisfied then all the other properties are also valid.

Our aim: give the conditions for the non-negativity preservation in 1D.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 35 / 54

Page 42: Qualitatively reliable numerical models of time-dependent problems · 2017-09-21 · Qualitatively reliable numerical models of time-dependent problems Istv an Farag o Farkas Mikl

Conditions for the non-negativity

For the finite difference method:

M = hI K =1

htridiag [−1, 2,−1],

For linear finite element method.

M =h

6tridiag [1, 4, 1], K =

1

htridiag [−1, 2,−1],

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 36 / 54

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Conditions:

X−11 ≥ 0 and X−1

1 X2 ≥ 0.

Using the special structure of the matrices, we can get strict conditions forthe number q := ∆t

h2 .

Two kinds of bounds:

uniform (valid for all space partitions)

depending on the dimension of the matrices (on h).

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 37 / 54

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Non-negativity of the finite difference scheme

For θ = 0: only under the condition q ≤ 0.5.

For θ = 1: no condition.For the values θ ∈ (0, 1)

For each N: iff under the condition

q ≤ 1

2(1− θ).

For each N ≥ 2: iff under the condition

q ≤2θ − 1 +

√1− θ(1− θ)

3θ(1− θ).

There exists an N0 such that the FDM is non-negativity preserving forall N ≥ N0 only under the condition

1

2(1− θ)< q <

1−√

1− θθ(1− θ)

.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 38 / 54

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Non-negativity of the linear finite element scheme

For each N: iff under the condition

1

6θ≤ q ≤ 1

3(1− θ).

There is a lower bound, too. Moreover, θ ∈ [1/3, 1] only.

For each N ≥ 2: iff under the condition

1

6θ≤ q ≤

3(2θ − 1) +√

9− 16θ(1− θ)

12θ(1− θ).

The necessary condition for existing an N0 number such that theFEM is non-negativity preserving for all N ≥ N0 can be obtained bysolving some algebraic equation.

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 39 / 54

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5. The Crank-Nicolson discretized heat conductionmodel

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 40 / 54

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The Crank-Nicolson discretized heat conduction models

θ = 0.5 in the general θ-method.

Discretization by the Crank-Nicolson method:

Mνn+1 − νn

∆t+ 0.5Kνn+1 + 0.5Kνn = η(n,θ) := 0.5ηn+1 + 0.5ηn. (11)

Farago Istvan (ELTE, BME, MTA-ELTE) Reliable Numerical Models for linear PDEs FMSZ 41 / 54

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History of the Crank-Nicolson method

The Crank-Nicolson method:

John Crank (1915 - 2006) originally worked in industry on themodelling and numerical solution of diffusion in polymers.

In 1943, working with Phyllis Nicolson (1917-1968) on finitedifference methods for the time dependent heat equation, heproposed the Crank-Nicolson method which has been incorporateduniversally in the solving of time-dependent problems since then.

Their first result on this method was published in 1947.

This is a special θ-method which corresponds to the choice θ = 0.5.

The stability function reads as

rCN(∆tM−1K) = (I− 0.5∆tM−1K)−1(I + 0.5∆tM−1K),

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Finite element method:

M = [Mij ]N×N , Mij =

∫Ω

φjφi dx =< φj , φi >,

K = [Kij ]N×N , Kij = B(φj , φi ) =< gradφj , gradφi >

M: the mass matrix, K: the stiffness matrix.

νni =< f (·, tn), φi > .

Approximation property:

νni :=

1

mes(φi )< f (·, tn), φi > .

Scaling by multiplying with the matrix D−1, where D = diag [mes(φi )].

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Conditions for the qualitative properties for theCN-method in 1D

For linear FEM in 1D on uniform mesh: D = diag [h],

M =h

6

4 1 . . . 0 1 0

1 4 1... 0 0

. . .. . .

. . ....

...... 1 4 1 0 00 . . . 1 4 0 1

,

K =1

h

2 −1 . . . 0 −1 0

−1 2 −1... 0 0

. . .. . .

. . ....

...... −1 2 −1 0 00 . . . −1 2 0 −1

.

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For FDM in 1D on unform mesh:

M = h

1 0 . . . 0 0 0

0 1 0... 0 0

. . .. . .

. . ....

...... 0 1 0 0 00 . . . 0 1 0 0

,

K =1

h

2 −1 . . . 0 −1 0

−1 2 −1... 0 0

. . .. . .

. . ....

...... −1 2 −1 0 00 . . . −1 2 0 −1

.

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Our aim: give the conditions for the non-negativity preservation in 1D.Conditions:

X−11 ≥ 0 and X−1

1 X2 ≥ 0.

X1 − X2 = K, hence DNP property ⇔ weak regular splitting of the(scaled) stiffness matrix

Using the special structure of the matrices, we can get strict conditions forthe number q := ∆t

h2 . (F., 1996)

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Our aim: give the conditions for the non-negativity preservation in 1D.Conditions:

X−11 ≥ 0 and X−1

1 X2 ≥ 0.

X1 − X2 = K, hence DNP property ⇔ weak regular splitting of the(scaled) stiffness matrix

Using the special structure of the matrices, we can get strict conditions forthe number q := ∆t

h2 . (F., 1996)

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Bounds for the Crank-Nicolson FDM on the uniform mesh:

For each N:q ≤ 1.

For each N ≥ 2:

q ≤ 2√3' 1.1547.

The necessary condition for existing an N0 number such that theFDM is non-negativity preserving for all N ≥ N0:

q < 2(2−√

2) ' 1.17157

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Bounds for the Crank-Nicolson FDM on the uniform mesh:

For each N:q ≤ 1.

For each N ≥ 2:

q ≤ 2√3' 1.1547.

The necessary condition for existing an N0 number such that theFDM is non-negativity preserving for all N ≥ N0:

q < 2(2−√

2) ' 1.17157

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Bounds for the Crank-Nicolson FDM on the uniform mesh:

For each N:q ≤ 1.

For each N ≥ 2:

q ≤ 2√3' 1.1547.

The necessary condition for existing an N0 number such that theFDM is non-negativity preserving for all N ≥ N0:

q < 2(2−√

2) ' 1.17157

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Bounds for the Crank-Nicolson linear FEM on the uniform mesh:

For each N: iff under the condition

1

3≤ q ≤ 2

3' 0.667.

For each N ≥ 2: iff under the condition

1

3≤ q ≤

√5

3' 0.745

The necessary condition for existing an N0 number such that theFEM is non-negativity preserving for all N ≥ N0:

1

3≤ q ≤' 0.748.

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Bounds for the Crank-Nicolson linear FEM on the uniform mesh:

For each N: iff under the condition

1

3≤ q ≤ 2

3' 0.667.

For each N ≥ 2: iff under the condition

1

3≤ q ≤

√5

3' 0.745

The necessary condition for existing an N0 number such that theFEM is non-negativity preserving for all N ≥ N0:

1

3≤ q ≤' 0.748.

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Bounds for the Crank-Nicolson linear FEM on the uniform mesh:

For each N: iff under the condition

1

3≤ q ≤ 2

3' 0.667.

For each N ≥ 2: iff under the condition

1

3≤ q ≤

√5

3' 0.745

The necessary condition for existing an N0 number such that theFEM is non-negativity preserving for all N ≥ N0:

1

3≤ q ≤' 0.748.

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The upper bound of the discrete maximum norm contractivity for theCrank-Nicolson method (Kraaijevanger,1992 and Horvath, 1996)

q ≤ 1.5

Hence, DMNC ; DNP.Qualitative behaviour of the operator “after the death”, i.e., in the caseq > 1.5:

‖νn‖∞ ≤ C∞‖ν0‖∞

Farago, Palencia (2004): 3 ≤ C∞ ≤' 4.32

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The upper bound of the discrete maximum norm contractivity for theCrank-Nicolson method (Kraaijevanger,1992 and Horvath, 1996)

q ≤ 1.5

Hence, DMNC ; DNP.

Qualitative behaviour of the operator “after the death”, i.e., in the caseq > 1.5:

‖νn‖∞ ≤ C∞‖ν0‖∞

Farago, Palencia (2004): 3 ≤ C∞ ≤' 4.32

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The upper bound of the discrete maximum norm contractivity for theCrank-Nicolson method (Kraaijevanger,1992 and Horvath, 1996)

q ≤ 1.5

Hence, DMNC ; DNP.Qualitative behaviour of the operator “after the death”, i.e., in the caseq > 1.5:

‖νn‖∞ ≤ C∞‖ν0‖∞

Farago, Palencia (2004): 3 ≤ C∞ ≤' 4.32

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Conditions for the qualitative properties of the CN-methodin IRd

Uniform regular simpletic mesh.

Our aim: give the conditions for the discrete weak maximum principle(DWMP) in IRd . (Problem to show the DNP property!)

Theorem

Let the basis functions fulfill the conditions φi ≥ 0 and∑N

i=1 φi ≡ 1.Then under the conditions

Kij ≤ 0, i 6= j , i = 1, ...,N, j = 1, ..., N,

Mij + 0.5∆t Kij ≤ 0, i 6= j , i = 1, ...,N, j = 1, ..., N,

Mii − 0.5∆t Kii ≥ 0, i = 1, ...,N

the discrete model has the DWMP property.

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Conditions for the qualitative properties in IRd -cont’d

The contributions to the mass matrix and stiffness matrix over the simplexT with the surface S in IRd :

Mij |T =1

(d + 1)(d + 2)measdT, i 6= j

Mii |T =2

(d + 1)(d + 2)measdT

Kij |T = −measd−1(Si) measd−1(Sj)

d2measdTcos γij , i 6= j

Kii |T =meas2d−1(Si)

d2measdT

where γij is the interior angle.

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Conditions for the qualitative properties in IRd -cont’d

For the regular simplex with the length h:

cos γij =1

d

measdT =hd

d!·√

d + 1

2d

The condition:h2

d + 2≤ ∆t ≤ 2h2

d(d + 2)

By this criteria the CN-method can be applied only when d ≤ 2. Ford = 2 the only possible choice: q = 0.25.

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Conditions for the qualitative properties in IRd -cont’d

For the regular simplex with the length h:

cos γij =1

d

measdT =hd

d!·√

d + 1

2d

The condition:h2

d + 2≤ ∆t ≤ 2h2

d(d + 2)

By this criteria the CN-method can be applied only when d ≤ 2. Ford = 2 the only possible choice: q = 0.25.

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Conditions for the qualitative properties in IRd -cont’d

For the regular simplex with the length h:

cos γij =1

d

measdT =hd

d!·√

d + 1

2d

The condition:h2

d + 2≤ ∆t ≤ 2h2

d(d + 2)

By this criteria the CN-method can be applied only when d ≤ 2. Ford = 2 the only possible choice: q = 0.25.

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Some references

I. F., R. Horvath [2006] SIAM Scientific Computing,

I. F., R. Horvath [2009] IMA Numerical Analysis

I. F., J. Karatson, S. Korotov.[2017] Math. and Comp. in Simulation

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Thank you for your attention

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