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annotate src/SimpleTimeSteppers/SimpleTimeSteppers.jl @ 862:a382942e5437 feature/subpackage_simple_timesteppers

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Add a few simple timesteppers
author Jonatan Werpers <jonatan@werpers.com>
date Tue, 18 Jan 2022 15:29:54 +0100
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1 """
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2 SimpleTimeSteppers
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3
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4 Implements a few simple time integration schemes useful for testing the accuracy of spatial discretisations.
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5 """
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6 module SimpleTimeSteppers
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7
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8 export rk4
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9 export rkn4
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10 export centered_2nd_order_fd
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11
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12 """
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13 rk4(F, vₙ, tₙ, Δtₙ)
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14
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15 Solves ``vₜ = F(v,t)`` using the standard 4th order RK-method.
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16 """
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17 function rk4(F, vₙ, tₙ, Δtₙ)
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18 k₁ = F(vₙ,tₙ)
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19 k₂ = F(vₙ + Δtₙ/2*k₁, tₙ + Δtₙ/2)
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20 k₃ = F(vₙ + Δtₙ/2*k₂, tₙ + Δtₙ/2)
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21 k₄ = F(vₙ + Δtₙ *k₃, tₙ + Δtₙ )
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22
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23 vₙ₊₁ = vₙ + (1/6)*(k₁+2*(k₂+k₃)+k₄)*Δtₙ
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24
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25 return vₙ₊₁
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26 end
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27
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28 """
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29 rkn4(F, vₙ, v̇ₙ, tₙ, Δtₙ)
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30
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31 Solves ``vₜₜ = F(v,v̇,t)`` using the Runge-Kutta-Nyström method based on the
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32 standard 4th order RK-method.
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33 """
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34 function rkn4(F, vₙ, v̇ₙ, tₙ, Δtₙ)
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35 k₁ = F(vₙ, v̇ₙ, tₙ )
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36 k₂ = F(vₙ + Δtₙ/2*v̇ₙ, v̇ₙ + Δtₙ/2*k₁, tₙ + Δtₙ/2)
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37 k₃ = F(vₙ + Δtₙ/2*v̇ₙ + Δtₙ^2/4*k₁, v̇ₙ + Δtₙ/2*k₂, tₙ + Δtₙ/2)
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38 k₄ = F(vₙ + Δtₙ *v̇ₙ + Δtₙ^2/2*k₂, v̇ₙ + Δtₙ *k₃, tₙ + Δtₙ )
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39
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40 vₙ₊₁ = vₙ + Δtₙ*v̇ₙ + Δtₙ^2*(1/6)*(k₁ + k₂ + k₃);
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41 v̇ₙ₊₁ = v̇ₙ + Δtₙ*(k₁ + 2*k₂ + 2*k₃ + k₄)/6;
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42
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43 return vₙ₊₁, v̇ₙ₊₁
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44 end
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45
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46 """
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47 centered_2nd_order_fd(F, vₙ, vₙ₋₁, Δtₙ)
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48
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49 Solves ``vₜₜ = F(v,t)`` using the 2nd order discretization
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50 ``(vₙ₊₁ - 2vₙ + vₙ₋₁)/Δt² = F(vₙ,tₙ)``.
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51 """
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52 function centered_2nd_order_fd(F, vₙ, vₙ₋₁, tₙ, Δt)
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53 vₙ₊₁ = Δt^2*F(vₙ,tₙ) + 2vₙ - vₙ₋₁
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54
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55 return vₙ₊₁, vₙ
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56 end
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57
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58 # TODO: Rewrite without allocations
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59
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60
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61 # TODO: Add euler forward and try a local error estimate for testing
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62 end # module