How to view this algorithm as a tridiagonal system? (Curve shortening flow)
I'm trying to create an implementation of a numerical method to approximate curve shortening flow, following "Computation of geometric partial differential equations and mean curvature flow" by Deckelnick, Dziuk and Elliott.
In one step of their algorithm, with $X_0^j$ defined (for $j=0,cdots,N$) they say to compute $X_j^{m+1}$ (again for $j=0,cdots,N$) from the tridiagonal systems $$(q_j^m+q_{j+1}^m)(X_j^{m+1}-X_j^m)-left(frac{X_{j+1}^{m+1}-X_j^{m+1}}{q_{j+1}^m}-frac{X_j^{m+1}-X_{j-1}^{m+1}}{q_j^m}right)=0,$$
where
$$q_j^m:=|X_j^m-X_{j-1}^m|.$$
I am having trouble understanding how this relates to the typically notated tridiagonal system so that I may solve it by the standard algorithms. Any tips on how to view/write it as such? (Sorry if this is a bit heavy or vague).
differential-geometry numerical-methods mean-curvature-flows
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
add a comment |
I'm trying to create an implementation of a numerical method to approximate curve shortening flow, following "Computation of geometric partial differential equations and mean curvature flow" by Deckelnick, Dziuk and Elliott.
In one step of their algorithm, with $X_0^j$ defined (for $j=0,cdots,N$) they say to compute $X_j^{m+1}$ (again for $j=0,cdots,N$) from the tridiagonal systems $$(q_j^m+q_{j+1}^m)(X_j^{m+1}-X_j^m)-left(frac{X_{j+1}^{m+1}-X_j^{m+1}}{q_{j+1}^m}-frac{X_j^{m+1}-X_{j-1}^{m+1}}{q_j^m}right)=0,$$
where
$$q_j^m:=|X_j^m-X_{j-1}^m|.$$
I am having trouble understanding how this relates to the typically notated tridiagonal system so that I may solve it by the standard algorithms. Any tips on how to view/write it as such? (Sorry if this is a bit heavy or vague).
differential-geometry numerical-methods mean-curvature-flows
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
add a comment |
I'm trying to create an implementation of a numerical method to approximate curve shortening flow, following "Computation of geometric partial differential equations and mean curvature flow" by Deckelnick, Dziuk and Elliott.
In one step of their algorithm, with $X_0^j$ defined (for $j=0,cdots,N$) they say to compute $X_j^{m+1}$ (again for $j=0,cdots,N$) from the tridiagonal systems $$(q_j^m+q_{j+1}^m)(X_j^{m+1}-X_j^m)-left(frac{X_{j+1}^{m+1}-X_j^{m+1}}{q_{j+1}^m}-frac{X_j^{m+1}-X_{j-1}^{m+1}}{q_j^m}right)=0,$$
where
$$q_j^m:=|X_j^m-X_{j-1}^m|.$$
I am having trouble understanding how this relates to the typically notated tridiagonal system so that I may solve it by the standard algorithms. Any tips on how to view/write it as such? (Sorry if this is a bit heavy or vague).
differential-geometry numerical-methods mean-curvature-flows
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
I'm trying to create an implementation of a numerical method to approximate curve shortening flow, following "Computation of geometric partial differential equations and mean curvature flow" by Deckelnick, Dziuk and Elliott.
In one step of their algorithm, with $X_0^j$ defined (for $j=0,cdots,N$) they say to compute $X_j^{m+1}$ (again for $j=0,cdots,N$) from the tridiagonal systems $$(q_j^m+q_{j+1}^m)(X_j^{m+1}-X_j^m)-left(frac{X_{j+1}^{m+1}-X_j^{m+1}}{q_{j+1}^m}-frac{X_j^{m+1}-X_{j-1}^{m+1}}{q_j^m}right)=0,$$
where
$$q_j^m:=|X_j^m-X_{j-1}^m|.$$
I am having trouble understanding how this relates to the typically notated tridiagonal system so that I may solve it by the standard algorithms. Any tips on how to view/write it as such? (Sorry if this is a bit heavy or vague).
differential-geometry numerical-methods mean-curvature-flows
differential-geometry numerical-methods mean-curvature-flows
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
asked Dec 1 '18 at 5:40
Finn ThompsonFinn Thompson
64
64
add a comment |
add a comment |
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