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Numerical methods for Maxwell’s equations Summer term 2014 Finite-difference method for the wave equation Tobias Jahnke Numerical methods for Maxwell’s equations Summer term 2014 Tobias Jahnke Karlsruhe Institute of Technology
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Finite-difference method for the wave equation
Tobias Jahnke
Summer term 2014
Numerical methods for Maxwell’s equations Summer term 2014
Model problem
Wave equation on [0, 2π]
∂2t u(t, x) = c2∂2xu(t, x) t ≥ 0, x ∈ (0, 2π)
u(0, x) = u0(x)
∂tu(0, x) = v0(x)
Periodic boundary conditions: u(t, 0) = u(t, 2π) Dirichlet boundary
conditions: u(t, 0) = a, u(t, 2π) = b
Finite-difference method (plus starting step)
un+1 j − 2unj + un−1
j
h2
Numerical methods for Maxwell’s equations Summer term 2014
Exact solution for different boundary conditions.
Tobias Jahnke Karlsruhe Institute of Technology
Numerical methods for Maxwell’s equations Summer term 2014
Exact solution for different boundary condition
Top: periodic boundary conditions Bottom: homogeneous Dirichlet
boundary conditions
Tobias Jahnke Karlsruhe Institute of Technology
Numerical methods for Maxwell’s equations Summer term 2014
Stability condition
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Stability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Same example with slightly larger time-step.
Tobias Jahnke Karlsruhe Institute of Technology
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Instability
0
0.2
0.4
0.6
0.8
1
x
Numerical methods for Maxwell’s equations Summer term 2014
Accuracy
Numerical methods for Maxwell’s equations Summer term 2014
Accuracy
If the solution is smooth and τ ≤ h c , then
max n=0,...,N
max j=1,...,m
1 c2
+ 1 ) h2
10 −3
10 −2
10 −1
10 −4
10 −3
10 −2
10 −1
10 0
error order 2
Maximal errors of approx- imations with different choices of h and
τ = r hc with fixed ratio r < 1.
If ε(h) = Chp, then log ( ε(h)
) = log(C ) + p log(h)
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
−1
−0.5
0
0.5
1
1.5
x
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
−1
−0.5
0
0.5
1
1.5
x
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
−1
−0.5
0
0.5
1
1.5
x
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
−1
−0.5
0
0.5
1
1.5
x
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
−1
−0.5
0
0.5
1
1.5
x
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
−1
−0.5
0
0.5
1
1.5
x
Numerical methods for Maxwell’s equations Summer term 2014
Dispersion
−1
−0.5
0
0.5
1
1.5
x
Numerical methods for Maxwell’s equations Summer term 2014
Finite-difference method for the wave equation
Tobias Jahnke
Summer term 2014
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