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  • 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