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Kirill
Kirill
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I'm going to guess that there isn't one. In the usual method of lines for $u_t +au_x=0$, you end up with a system of ODEs of the form $$ u_t = Au. $$

So the restriction on the time step comes from (1) requiring that eigenvalues of $A$ lie in the domain of stability of whatever time-stepping method you use (e.g., if $A$ is skew-symmetric, with imaginary eigenvalues, then forward Euler is always unstable), and (2) from minimizing error (if both space and time discretizations are second-order, then $\delta t=\delta x$).

So based on this reasoning, I'd say no, because there is too much freedom in how you can pick the r.h.s. matrix and the time-stepping method, and the result has less to do specifically with the order of the method, and more to do with other properties, mainly $A$'s spectrum. Of course this doesn't quite rule out such a formula, but my understanding is that the stability analysis has to be done case-by-case.

I'm going to guess that there isn't one. In the usual method of lines for $u_t +au_x=0$, you end up with a system of ODEs of the form $$ u_t = Au. $$

So the restriction on the time step comes from (1) requiring that eigenvalues of $A$ lie in the domain of stability of whatever time-stepping method you use (e.g., if $A$ is skew-symmetric, with imaginary eigenvalues, then forward Euler is always unstable), and (2) from minimizing error (if both space and time discretizations are second-order, then $\delta t=\delta x$).

So based on this reasoning, I'd say no, because there is too much freedom in how you can pick the r.h.s. matrix and the time-stepping method, and the result has less to do specifically with the order of the method, and more to do with other properties. Of course this doesn't quite rule out such a formula, but my understanding is that the stability analysis has to be done case-by-case.

I'm going to guess that there isn't one. In the usual method of lines for $u_t +au_x=0$, you end up with a system of ODEs of the form $$ u_t = Au. $$

So the restriction on the time step comes from (1) requiring that eigenvalues of $A$ lie in the domain of stability of whatever time-stepping method you use (e.g., if $A$ is skew-symmetric, with imaginary eigenvalues, then forward Euler is always unstable), and (2) from minimizing error (if both space and time discretizations are second-order, then $\delta t=\delta x$).

So based on this reasoning, I'd say no, because there is too much freedom in how you can pick the r.h.s. matrix and the time-stepping method, and the result has less to do specifically with the order of the method, and more to do with other properties, mainly $A$'s spectrum. Of course this doesn't quite rule out such a formula, but my understanding is that the stability analysis has to be done case-by-case.

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Kirill
Kirill
  • 11.5k
  • 2
  • 27
  • 51

I'm going to guess that there isn't one. In the usual method of lines for $u_t +au_x=0$, you end up with a system of ODEs of the form $$ u_t = Au. $$

So the restriction on the time step comes from (1) requiring that eigenvalues of $A$ lie in the domain of stability of whatever time-stepping method you use (e.g., if $A$ is skew-symmetric, with imaginary eigenvalues, then forward Euler is always unstable), and (2) from minimizing error (if both space and time discretizations are second-order, then $\delta t=\delta x$).

So based on this reasoning, I'd say no, because there is too much freedom in how you can pick the r.h.s. matrix and the time-stepping method, and the result has less to do specifically with the order of the method, and more to do with other properties. Of course this doesn't quite rule out such a formula, but my understanding is that the stability analysis has to be done case-by-case.