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jverzani
2025-04-16 14:31:16 -04:00
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quarto/ODEs/solve.qmd
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@@ -118,10 +118,10 @@ function solve(prob::Problem, alg::EulerMethod)
end
```
The post has a more elegant means to unpack the parameters from the structures, but for each of the above, the parameters are unpacked, and then the corresponding algorithm employed. As of version `v1.7` of `Julia`, the syntax `(;g,y0,v0,tspan) = prob` could also be employed.
The post has a more elegant means to unpack the parameters from the structures, but for each of the above, the parameters are unpacked using the dot notation for `getproperty`, and then the corresponding algorithm employed. As of version `v1.7` of `Julia`, the syntax `(;g,y0,v0,tspan) = prob` could also have been employed.
The exact formulas, `y(t) = y0 + v0*(t - t0) - g*(t - t0)^2/2` and `v(t) = v0 - g*(t - t0)`, follow from well-known physics formulas. Each answer is wrapped in a `Solution` type so that the answers found can be easily extracted in a uniform manner.
The exact answers, `y(t) = y0 + v0*(t - t0) - g*(t - t0)^2/2` and `v(t) = v0 - g*(t - t0)`, follow from well-known physics formulas for constant-acceleration motion. Each answer is wrapped in a `Solution` type so that the answers found can be easily extracted in a uniform manner.
For example, plots of each can be obtained through:
@@ -138,7 +138,9 @@ plot!(sol_exact.t, sol_exact.y; label="exact solution", ls=:auto)
title!("On the Earth"; xlabel="t", legend=:bottomleft)
```
Following the post, since the time step `dt = 0.1` is not small enough, the error of the Euler method is rather large. Next we change the algorithm parameter, `dt`, to be smaller:
Following the post, since the time step `dt = 0.1` is not small enough, the error of the Euler method is readily identified.
Next we change the algorithm parameter, `dt`, to be smaller:
```{julia}
@@ -155,7 +157,7 @@ title!("On the Earth"; xlabel="t", legend=:bottomleft)
It is worth noting that only the first line is modified, and only the method requires modification.
Were the moon to be considered, the gravitational constant would need adjustment. This parameter is part of the problem, not the solution algorithm.
Were the moon to be considered, the gravitational constant would need adjustment. This parameter is a property of the problem, not the solution algorithm, as `dt` is.
Such adjustments are made by passing different values to the `Problem` constructor:
@@ -175,7 +177,9 @@ title!("On the Moon"; xlabel="t", legend=:bottomleft)
The code above also adjusts the time span in addition to the graviational constant. The algorithm for exact formula is set to use the `dt` value used in the `euler` formula, for easier comparison. Otherwise, outside of the labels, the patterns are the same. Only those things that need changing are changed, the rest comes from defaults.
The above shows the benefits of using a common interface. Next, the post illustrates how *other* authors could extend this code, simply by adding a *new* `solve` method. For example,
The above shows the benefits of using a common interface.
Next, the post illustrates how *other* authors could extend this code, simply by adding a *new* `solve` method. For example, a sympletic method conserves a quantity, so can track long-term evolution without drift.
```{julia}