Finite Element in space, finite difference in time, stability analysisBackward Euler for a system of equationsVon Neumann Stability AnalysisMixed finite element methods and time-dependenceInterval arithmetic for finite difference error boundsFinite Difference Method Stability with diffusion equationStability analysis for ODEs with non constant inputsWhat is the difference between a stability condition and a CFL condition?Explicit finite difference scheme for discontinued mediaUsing quadrature for the computation of integrals occuring in the finite element methodHelp with an inequality, for von Neumann stability analysis.
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Finite Element in space, finite difference in time, stability analysis
Backward Euler for a system of equationsVon Neumann Stability AnalysisMixed finite element methods and time-dependenceInterval arithmetic for finite difference error boundsFinite Difference Method Stability with diffusion equationStability analysis for ODEs with non constant inputsWhat is the difference between a stability condition and a CFL condition?Explicit finite difference scheme for discontinued mediaUsing quadrature for the computation of integrals occuring in the finite element methodHelp with an inequality, for von Neumann stability analysis.
$begingroup$
I was wondering how to do stability analysis for the classical wave equation if we discretize in space with finite element and step forward in time using some explicit scheme like for example leapfrog or Euler forward.
A reference or example of such a calculation would be much appreciated.
calculus numerical-methods stability-in-odes
asked Mar 11 at 19:18
JHadamardJHadamard
84
$endgroup$
add a comment |
$begingroup$
I was wondering how to do stability analysis for the classical wave equation if we discretize in space with finite element and step forward in time using some explicit scheme like for example leapfrog or Euler forward.
A reference or example of such a calculation would be much appreciated.
calculus numerical-methods stability-in-odes
asked Mar 11 at 19:18
JHadamardJHadamard
84
$endgroup$
add a comment |
$begingroup$
I was wondering how to do stability analysis for the classical wave equation if we discretize in space with finite element and step forward in time using some explicit scheme like for example leapfrog or Euler forward.
A reference or example of such a calculation would be much appreciated.
calculus numerical-methods stability-in-odes
asked Mar 11 at 19:18
JHadamardJHadamard
84
$endgroup$
I was wondering how to do stability analysis for the classical wave equation if we discretize in space with finite element and step forward in time using some explicit scheme like for example leapfrog or Euler forward.
A reference or example of such a calculation would be much appreciated.
calculus numerical-methods stability-in-odes
calculus numerical-methods stability-in-odes
asked Mar 11 at 19:18
JHadamardJHadamard
84
asked Mar 11 at 19:18
JHadamardJHadamard
84
asked Mar 11 at 19:18
JHadamardJHadamard
84
asked Mar 11 at 19:18
JHadamardJHadamard
84
asked Mar 11 at 19:18
JHadamardJHadamard
84
84
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I assume that the starting level with discretizing stuff is "zero". The first thing most people would try would be something like this. The PDE is
$$ Delta u + u_tt = 0 text in Omega$$
with some appropriate initial and boundary conditions. The most obvious approach would be to write $v = u_t$, and get a system of equations
beginalign*
Delta u + v_t &= 0, \
u_t - v &= 0.
endalign*
Then, the most straightforward discretization would be to use the same finite element basis functions $phi_i$ for $u$ and $v$, leading to a system
beginalign*
Ku + M dot v &= 0, \
M dot u - M v &= 0,
endalign*
where $K_ij = int nabla phi_i cdot nabla phi_j , dx$ is the stiffness matrix, and $M_ij = int phi_i phi_j , dx$ the mass matrix. Rearranging the previous linear system, you get
$$
beginbmatrix dot u \ dot v endbmatrix =
beginbmatrix 0 & I \ -M^-1 K & 0 endbmatrix beginbmatrix u \ v endbmatrix = J beginbmatrix u \ v endbmatrix
$$
Assuming, e.g., boundary condition $u=0$ on $partial Omega$, the matrix $-M^-1 K$ is negative definite, so the matrix $J$ looks very much like $beginbmatrix 0 & 1 \ -1 & 0 endbmatrix$, that is, the ODE system looks like a harmonic oscillator.
Consider a simple harmonic oscillator
$$
beginbmatrix dot q \ dot p endbmatrix =
beginbmatrix 0 & 1 \ -1 & 0 endbmatrix beginbmatrix q \ p endbmatrix,
$$
which has the energy $W = p^2 + q^2$ as a conserved quantity. It is a straightforward computation to show that the forward Euler increases the energy each time step, whereas the leap frog conserves energy exactly each time step. By doing a similar computation for the above finite element-ODE system, you should find a similar result (try to identify the conserved quantity $W$ first).
This is just to get started. In real life, more advanced schemes are used. For example, the first Google hit for "wave equation FEM" is https://hal.archives-ouvertes.fr/hal-01443184/document
answered Mar 11 at 20:35
user635750user635750
363
$endgroup$
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I assume that the starting level with discretizing stuff is "zero". The first thing most people would try would be something like this. The PDE is
$$ Delta u + u_tt = 0 text in Omega$$
with some appropriate initial and boundary conditions. The most obvious approach would be to write $v = u_t$, and get a system of equations
beginalign*
Delta u + v_t &= 0, \
u_t - v &= 0.
endalign*
Then, the most straightforward discretization would be to use the same finite element basis functions $phi_i$ for $u$ and $v$, leading to a system
beginalign*
Ku + M dot v &= 0, \
M dot u - M v &= 0,
endalign*
where $K_ij = int nabla phi_i cdot nabla phi_j , dx$ is the stiffness matrix, and $M_ij = int phi_i phi_j , dx$ the mass matrix. Rearranging the previous linear system, you get
$$
beginbmatrix dot u \ dot v endbmatrix =
beginbmatrix 0 & I \ -M^-1 K & 0 endbmatrix beginbmatrix u \ v endbmatrix = J beginbmatrix u \ v endbmatrix
$$
Assuming, e.g., boundary condition $u=0$ on $partial Omega$, the matrix $-M^-1 K$ is negative definite, so the matrix $J$ looks very much like $beginbmatrix 0 & 1 \ -1 & 0 endbmatrix$, that is, the ODE system looks like a harmonic oscillator.
Consider a simple harmonic oscillator
$$
beginbmatrix dot q \ dot p endbmatrix =
beginbmatrix 0 & 1 \ -1 & 0 endbmatrix beginbmatrix q \ p endbmatrix,
$$
which has the energy $W = p^2 + q^2$ as a conserved quantity. It is a straightforward computation to show that the forward Euler increases the energy each time step, whereas the leap frog conserves energy exactly each time step. By doing a similar computation for the above finite element-ODE system, you should find a similar result (try to identify the conserved quantity $W$ first).
This is just to get started. In real life, more advanced schemes are used. For example, the first Google hit for "wave equation FEM" is https://hal.archives-ouvertes.fr/hal-01443184/document
answered Mar 11 at 20:35
user635750user635750
363
$endgroup$
add a comment |
$begingroup$
I assume that the starting level with discretizing stuff is "zero". The first thing most people would try would be something like this. The PDE is
$$ Delta u + u_tt = 0 text in Omega$$
with some appropriate initial and boundary conditions. The most obvious approach would be to write $v = u_t$, and get a system of equations
beginalign*
Delta u + v_t &= 0, \
u_t - v &= 0.
endalign*
Then, the most straightforward discretization would be to use the same finite element basis functions $phi_i$ for $u$ and $v$, leading to a system
beginalign*
Ku + M dot v &= 0, \
M dot u - M v &= 0,
endalign*
where $K_ij = int nabla phi_i cdot nabla phi_j , dx$ is the stiffness matrix, and $M_ij = int phi_i phi_j , dx$ the mass matrix. Rearranging the previous linear system, you get
$$
beginbmatrix dot u \ dot v endbmatrix =
beginbmatrix 0 & I \ -M^-1 K & 0 endbmatrix beginbmatrix u \ v endbmatrix = J beginbmatrix u \ v endbmatrix
$$
Assuming, e.g., boundary condition $u=0$ on $partial Omega$, the matrix $-M^-1 K$ is negative definite, so the matrix $J$ looks very much like $beginbmatrix 0 & 1 \ -1 & 0 endbmatrix$, that is, the ODE system looks like a harmonic oscillator.
Consider a simple harmonic oscillator
$$
beginbmatrix dot q \ dot p endbmatrix =
beginbmatrix 0 & 1 \ -1 & 0 endbmatrix beginbmatrix q \ p endbmatrix,
$$
which has the energy $W = p^2 + q^2$ as a conserved quantity. It is a straightforward computation to show that the forward Euler increases the energy each time step, whereas the leap frog conserves energy exactly each time step. By doing a similar computation for the above finite element-ODE system, you should find a similar result (try to identify the conserved quantity $W$ first).
This is just to get started. In real life, more advanced schemes are used. For example, the first Google hit for "wave equation FEM" is https://hal.archives-ouvertes.fr/hal-01443184/document
answered Mar 11 at 20:35
user635750user635750
363
$endgroup$
add a comment |
$begingroup$
I assume that the starting level with discretizing stuff is "zero". The first thing most people would try would be something like this. The PDE is
$$ Delta u + u_tt = 0 text in Omega$$
with some appropriate initial and boundary conditions. The most obvious approach would be to write $v = u_t$, and get a system of equations
beginalign*
Delta u + v_t &= 0, \
u_t - v &= 0.
endalign*
Then, the most straightforward discretization would be to use the same finite element basis functions $phi_i$ for $u$ and $v$, leading to a system
beginalign*
Ku + M dot v &= 0, \
M dot u - M v &= 0,
endalign*
where $K_ij = int nabla phi_i cdot nabla phi_j , dx$ is the stiffness matrix, and $M_ij = int phi_i phi_j , dx$ the mass matrix. Rearranging the previous linear system, you get
$$
beginbmatrix dot u \ dot v endbmatrix =
beginbmatrix 0 & I \ -M^-1 K & 0 endbmatrix beginbmatrix u \ v endbmatrix = J beginbmatrix u \ v endbmatrix
$$
Assuming, e.g., boundary condition $u=0$ on $partial Omega$, the matrix $-M^-1 K$ is negative definite, so the matrix $J$ looks very much like $beginbmatrix 0 & 1 \ -1 & 0 endbmatrix$, that is, the ODE system looks like a harmonic oscillator.
Consider a simple harmonic oscillator
$$
beginbmatrix dot q \ dot p endbmatrix =
beginbmatrix 0 & 1 \ -1 & 0 endbmatrix beginbmatrix q \ p endbmatrix,
$$
which has the energy $W = p^2 + q^2$ as a conserved quantity. It is a straightforward computation to show that the forward Euler increases the energy each time step, whereas the leap frog conserves energy exactly each time step. By doing a similar computation for the above finite element-ODE system, you should find a similar result (try to identify the conserved quantity $W$ first).
This is just to get started. In real life, more advanced schemes are used. For example, the first Google hit for "wave equation FEM" is https://hal.archives-ouvertes.fr/hal-01443184/document
answered Mar 11 at 20:35
user635750user635750
363
$endgroup$
I assume that the starting level with discretizing stuff is "zero". The first thing most people would try would be something like this. The PDE is
$$ Delta u + u_tt = 0 text in Omega$$
with some appropriate initial and boundary conditions. The most obvious approach would be to write $v = u_t$, and get a system of equations
beginalign*
Delta u + v_t &= 0, \
u_t - v &= 0.
endalign*
Then, the most straightforward discretization would be to use the same finite element basis functions $phi_i$ for $u$ and $v$, leading to a system
beginalign*
Ku + M dot v &= 0, \
M dot u - M v &= 0,
endalign*
where $K_ij = int nabla phi_i cdot nabla phi_j , dx$ is the stiffness matrix, and $M_ij = int phi_i phi_j , dx$ the mass matrix. Rearranging the previous linear system, you get
$$
beginbmatrix dot u \ dot v endbmatrix =
beginbmatrix 0 & I \ -M^-1 K & 0 endbmatrix beginbmatrix u \ v endbmatrix = J beginbmatrix u \ v endbmatrix
$$
Assuming, e.g., boundary condition $u=0$ on $partial Omega$, the matrix $-M^-1 K$ is negative definite, so the matrix $J$ looks very much like $beginbmatrix 0 & 1 \ -1 & 0 endbmatrix$, that is, the ODE system looks like a harmonic oscillator.
Consider a simple harmonic oscillator
$$
beginbmatrix dot q \ dot p endbmatrix =
beginbmatrix 0 & 1 \ -1 & 0 endbmatrix beginbmatrix q \ p endbmatrix,
$$
which has the energy $W = p^2 + q^2$ as a conserved quantity. It is a straightforward computation to show that the forward Euler increases the energy each time step, whereas the leap frog conserves energy exactly each time step. By doing a similar computation for the above finite element-ODE system, you should find a similar result (try to identify the conserved quantity $W$ first).
This is just to get started. In real life, more advanced schemes are used. For example, the first Google hit for "wave equation FEM" is https://hal.archives-ouvertes.fr/hal-01443184/document
answered Mar 11 at 20:35
user635750user635750
363
answered Mar 11 at 20:35
user635750user635750
363
answered Mar 11 at 20:35
user635750user635750
363
answered Mar 11 at 20:35
user635750user635750
363
363
add a comment |
add a comment |
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