Testing convergence rates of numerical solution with no known solution
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I am trying to test the accuracy of my code to a PDE with no numerical solution. I am using a Backwards time centred space finite difference method. I am calculating the error using;
$frac{u_{4h} - u_{2h}}{u_{2h} - u_{h}} = 2^p + mathcal{O}(h)$,
where $h$ is the step size, $p$ is the order of the method and $u$ is my approximation. I am calculating the error using different spatial/time steps and calculating p. My scheme should be $mathcal{O}(Delta t, Delta x^2)$ and so I expect $p$ to be $1$ and $2$ respectively - this isn't what I get.
But my question is what is the best way to test the numerical accuracy of a solution to without any know analytic solution? I am aware of the method of manufactured solution, but wanting to use something more similar to above. Ideally, I could plot the error against $Delta t$ and $Delta x^2$ on a log log plot where I would get a straight line.
I am quite new to numerical methods so there may be a very obvious answer, but I have tried searching for this before asking my question here.
convergence numerical-methods finite-differences truncation-error
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
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add a comment |
$begingroup$
I am trying to test the accuracy of my code to a PDE with no numerical solution. I am using a Backwards time centred space finite difference method. I am calculating the error using;
$frac{u_{4h} - u_{2h}}{u_{2h} - u_{h}} = 2^p + mathcal{O}(h)$,
where $h$ is the step size, $p$ is the order of the method and $u$ is my approximation. I am calculating the error using different spatial/time steps and calculating p. My scheme should be $mathcal{O}(Delta t, Delta x^2)$ and so I expect $p$ to be $1$ and $2$ respectively - this isn't what I get.
But my question is what is the best way to test the numerical accuracy of a solution to without any know analytic solution? I am aware of the method of manufactured solution, but wanting to use something more similar to above. Ideally, I could plot the error against $Delta t$ and $Delta x^2$ on a log log plot where I would get a straight line.
I am quite new to numerical methods so there may be a very obvious answer, but I have tried searching for this before asking my question here.
convergence numerical-methods finite-differences truncation-error
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
$endgroup$
$begingroup$
What PDE are you solving?
$endgroup$
– ekkilop
Dec 28 '18 at 22:43
add a comment |
$begingroup$
I am trying to test the accuracy of my code to a PDE with no numerical solution. I am using a Backwards time centred space finite difference method. I am calculating the error using;
$frac{u_{4h} - u_{2h}}{u_{2h} - u_{h}} = 2^p + mathcal{O}(h)$,
where $h$ is the step size, $p$ is the order of the method and $u$ is my approximation. I am calculating the error using different spatial/time steps and calculating p. My scheme should be $mathcal{O}(Delta t, Delta x^2)$ and so I expect $p$ to be $1$ and $2$ respectively - this isn't what I get.
But my question is what is the best way to test the numerical accuracy of a solution to without any know analytic solution? I am aware of the method of manufactured solution, but wanting to use something more similar to above. Ideally, I could plot the error against $Delta t$ and $Delta x^2$ on a log log plot where I would get a straight line.
I am quite new to numerical methods so there may be a very obvious answer, but I have tried searching for this before asking my question here.
convergence numerical-methods finite-differences truncation-error
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
$endgroup$
I am trying to test the accuracy of my code to a PDE with no numerical solution. I am using a Backwards time centred space finite difference method. I am calculating the error using;
$frac{u_{4h} - u_{2h}}{u_{2h} - u_{h}} = 2^p + mathcal{O}(h)$,
where $h$ is the step size, $p$ is the order of the method and $u$ is my approximation. I am calculating the error using different spatial/time steps and calculating p. My scheme should be $mathcal{O}(Delta t, Delta x^2)$ and so I expect $p$ to be $1$ and $2$ respectively - this isn't what I get.
But my question is what is the best way to test the numerical accuracy of a solution to without any know analytic solution? I am aware of the method of manufactured solution, but wanting to use something more similar to above. Ideally, I could plot the error against $Delta t$ and $Delta x^2$ on a log log plot where I would get a straight line.
I am quite new to numerical methods so there may be a very obvious answer, but I have tried searching for this before asking my question here.
convergence numerical-methods finite-differences truncation-error
convergence numerical-methods finite-differences truncation-error
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
asked Dec 13 '18 at 13:20
Patrick LewisPatrick Lewis
112
112
$begingroup$
What PDE are you solving?
$endgroup$
– ekkilop
Dec 28 '18 at 22:43
add a comment |
$begingroup$
What PDE are you solving?
$endgroup$
– ekkilop
Dec 28 '18 at 22:43
$begingroup$
What PDE are you solving?
$endgroup$
– ekkilop
Dec 28 '18 at 22:43
$begingroup$
What PDE are you solving?
$endgroup$
– ekkilop
Dec 28 '18 at 22:43
add a comment |
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$begingroup$
What PDE are you solving?
$endgroup$
– ekkilop
Dec 28 '18 at 22:43