Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

Numerical methods for stiff Ordinary differential equations. Application to the

Finite Element Method (FEM)

Elisabete Alberdi Celaya

EUIT de Minas y Obras Públicas UPV/EHU, Paseo Rafael Moreno Pitxitxi 2, 48013 Bilbao (Vizcaya)

April 7, 2013

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

Index

1 Introduction

2 Numerical methods for first order ODEs

3 Changing the predictor in EBDF and MEBDF methods

4 Linear multistep methods for second order ODEs

5 BDF-α method

6 Object Oriented Programming methodology

7 Results

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

0

1

N i (x

j )=δ

ij

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

0

1

N i (x

j )=δ

ij

FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

0

1

N i (x

j )=δ

ij

FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)

Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

0

1

N i (x

j )=δ

ij

FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)

Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1

Weak formulation: ∫ L0 Ni ρcput dx = ∫ L

0 N ′

i ku h x dx, i = 2, ..., n − 1

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

0

1

N i (x

j )=δ

ij

FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)

Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1

Weak formulation: ∫ L0 Ni ρcput dx = ∫ L

0 N ′

i ku h x dx, i = 2, ..., n − 1

Ordinary Differential Equations System:

∑ n−1 j=2



     

∫ L

0 ρcpNi (x)Nj (x)dx

︸ ︷︷ ︸

mij



     

d ′j (t) = − ∑ n−1

j=2



     

∫ L

0 kN′i (x)N

′

j (x)dx

︸ ︷︷ ︸

kij



     

dj (t), i, j = 2, ..., n − 1

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

0

1

N i (x

j )=δ

ij

FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)

Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1

Weak formulation: ∫ L0 Ni ρcput dx = ∫ L

0 N ′

i ku h x dx, i = 2, ..., n − 1

Ordinary Differential Equations System:

∑ n−1 j=2



     

∫ L

0 ρcpNi (x)Nj (x)dx

︸ ︷︷ ︸

mij



     

d ′j (t) = − ∑ n−1

j=2



     

∫ L

0 kN′i (x)N

′

j (x)dx

︸ ︷︷ ︸

kij



     

dj (t), i, j = 2, ..., n − 1

DIFFUSION EQUATION:

{

Md′(t) = α2K d(t), IC : d0i = g(x i), ∀i ∈ ηd

Numerical methods for stiff ODEs

Elisabete Alberdi Celaya

Introduction

First order ODEs

Changing EBDFs and MEBDFs

LMS for second order ODEs

BDF-α method

OOP methodology

Results

A FEM application to the 1D linear diffusion equation

PDEs→ FEM approximation

� Difussion:







ρcput = kuxx , ∀x ∈ [0, L], t ∈ [0,∞)

BC : u(0, t) = 0 = u(L, t)

IC : u(x , 0) = g(x), ∀x ∈ [0, L]

0

1

N i (x

j )=δ

ij

FEM approximation: u(x, t) ≈ uh(x, t) = ∑ n−1j=2 dj (t)Nj (x)

Orthogonality of the residual: ∫ L0 Ni (ρcput − kuxx )dx = 0, i = 2, ..., n − 1

Weak formulation: ∫ L0 Ni ρcput dx = ∫ L

0 N ′

i ku h x dx, i = 2, ..., n − 1

Ordinary Differential Equations System:

∑ n−1 j=2



     

∫ L

0 ρcpNi (x)Nj (x)dx

︸ ︷︷ ︸

mij



     

d ′j (t) = − ∑ n−1

j=2



     

∫ L

0 kN′i (x)N

′

j (x)dx

︸ ︷︷ ︸

kij



     

dj (t), i, j = 2, ..., n − 1

DIFFUSION EQUA