# Computer Programs (ch03:program-design)= ## Program Design ::::::{grid} ::::{grid-item-card} :columns: 4 ```{mermaid} :align: center flowchart TD id1([Start]) --> id2[Define IVP] --> id3[Produce Starting Values] --> id4[Apply Multistep Method] --> id5[/Output Result/] --> id6([END]) ``` :::: ::::{grid-item} :columns: 8 ```{prf:algorithm} Linear Multistep Method for IVPs :label: algorithm-LMM **Define** the IVP - The ODE $y'=f(t, y)$ - The interval of definition $I=[\text{tStart}, \text{tEnd}]$ - Number of points $N$ on $I$ - Step size $h=(\text{tEnd}-\text{tStart})/(N-1)$ - Initial values: $t_0$, $y_0$, $f_0$ **Produce** starting values from step $1$ to step $k-1$: - $y_1$, $y_2$, ..., $y_{k-1}$ - $f_1$, $f_2$, ..., $f_{k-1}$ **Apply** the linear multistep method for the remaining steps $$ \sum_{i=0}^{k} \alpha_i y_{j-k+i} = h\sum_{i=0}^{k}\beta_i f_{j-k+i}, \quad j=k, \ldots, N-1 $$ **Output** the result $\{y_j\}_{j=0}^{N-1}$ ``` :::: :::::: (ch03:example-program)= ## Example Programs In order to illustrate the predictor-corrector methods computationally, three programs with the corresponding function definitions and the solution tables are appended below: - **[](ch03:program1)** - solves the problem using the exact solution to find the starting values, followed by the $4^\text{th}$order ABM method for the remaining integration points. - **[](ch03:program2)** - implements the classic $4^{th}$--order Runge--Kutta method for computing the starting values, followed by the $4^{th}$order ABM method for the remaining integration points. - **[](ch03:program3)** - uses the ODE solver ode113 (based on non-stiff ABM methods) from the ODE solver routines built-in Matlab. This will be discussed in detail using Matlab documentation attached. Run these programs to check your results for {prf:ref}`example-3.2`. (ch03:program1)= ### Program 1 (Matlab and Python) Using the **analytical solution** to produce starting values. :::::{tab-set} ::::{tab-item} MATLAB The following code has been tested under **Matlab 2025** ```{literalinclude} /codes/ch2_prog1.m :linenos: True :language: matlab :emphasize-lines: 28,30 ``` The output of this code is: ```none x yp y F y_ex AbsError 0.00 0.0000000 1.0000000 0.0000000 1.0000000 0.00000000 0.10 0.0000000 1.0048374 0.0951626 1.0048374 0.00000000 0.20 0.0000000 1.0187308 0.1812692 1.0187308 0.00000000 0.30 0.0000000 1.0408182 0.2591818 1.0408182 0.00000000 0.40 1.0703229 1.0703197 0.3296803 1.0703200 0.00000031 0.50 1.1065330 1.1065301 0.3934699 1.1065307 0.00000056 0.60 1.1488135 1.1488109 0.4511891 1.1488116 0.00000075 0.70 1.1965868 1.1965844 0.5034156 1.1965853 0.00000091 0.80 1.2493301 1.2493279 0.5506721 1.2493290 0.00000103 0.90 1.3065705 1.3065685 0.5934315 1.3065697 0.00000111 1.00 1.3678800 1.3678783 0.6321217 1.3678794 0.00000117 ``` :::: ::::{tab-item} Python ```{literalinclude} /codes/ch2_prog1.py :linenos: True :language: python :emphasize-lines: 34,37 ``` The output of this code is: ```none t yp y F y_ex AbsError 0.00 0.0000000 1.0000000 0.0000000 1.0000000 0.00000000 0.10 0.0000000 1.0048374 0.0951626 1.0048374 0.00000000 0.20 0.0000000 1.0187308 0.1812692 1.0187308 0.00000000 0.30 0.0000000 1.0408182 0.2591818 1.0408182 0.00000000 0.40 1.0703229 1.0703197 0.3296803 1.0703200 0.00000031 0.50 1.1065330 1.1065301 0.3934699 1.1065307 0.00000056 0.60 1.1488135 1.1488109 0.4511891 1.1488116 0.00000075 0.70 1.1965868 1.1965844 0.5034156 1.1965853 0.00000091 0.80 1.2493301 1.2493279 0.5506721 1.2493290 0.00000103 0.90 1.3065705 1.3065685 0.5934315 1.3065697 0.00000111 1.00 1.3678800 1.3678783 0.6321217 1.3678794 0.00000117 ``` :::: ::::: (ch03:program2)= ### Program 2 (Matlab and Python) Using the **RK4** method to produce starting values. :::::{tab-set} ::::{tab-item} MATLAB The following code has been tested under **Matlab 2025** ```{literalinclude} /codes/ch2_prog2.m :linenos: True :language: matlab :emphasize-lines: 32,34 ``` Output: ```none x yp y F y_ex AbsError 0.00 0.0000000 1.0000000 0.0000000 1.0000000 0.00000000 0.10 0.0000000 1.0048375 0.0951625 1.0048374 0.00000008 0.20 0.0000000 1.0187309 0.1812691 1.0187308 0.00000015 0.30 0.0000000 1.0408184 0.2591816 1.0408182 0.00000020 0.40 1.0703231 1.0703199 0.3296801 1.0703200 0.00000013 0.50 1.1065332 1.1065303 0.3934697 1.1065307 0.00000039 0.60 1.1488136 1.1488110 0.4511890 1.1488116 0.00000060 0.70 1.1965869 1.1965845 0.5034155 1.1965853 0.00000077 0.80 1.2493302 1.2493281 0.5506719 1.2493290 0.00000090 0.90 1.3065706 1.3065687 0.5934313 1.3065697 0.00000100 1.00 1.3678801 1.3678784 0.6321216 1.3678794 0.00000108 ``` :::: ::::{tab-item} Python ```{literalinclude} /codes/ch2_prog2.py :linenos: True :language: python :emphasize-lines: 38,41 ``` Output: ```none t yp y F y_ex AbsError 0.00 0.0000000 1.0000000 0.0000000 1.0000000 0.00000000 0.10 0.0000000 1.0048375 0.0951625 1.0048374 0.00000008 0.20 0.0000000 1.0187309 0.1812691 1.0187308 0.00000015 0.30 0.0000000 1.0408184 0.2591816 1.0408182 0.00000020 0.40 1.0703231 1.0703199 0.3296801 1.0703200 0.00000013 0.50 1.1065332 1.1065303 0.3934697 1.1065307 0.00000039 0.60 1.1488136 1.1488110 0.4511890 1.1488116 0.00000060 0.70 1.1965869 1.1965845 0.5034155 1.1965853 0.00000077 0.80 1.2493302 1.2493281 0.5506719 1.2493290 0.00000090 0.90 1.3065706 1.3065687 0.5934313 1.3065697 0.00000100 1.00 1.3678801 1.3678784 0.6321216 1.3678794 0.00000108 ``` :::: ::::: (ch03:program3)= ### Program 3 (Matlab only) The following code has been tested under **Matlab 2025** :::{literalinclude} /codes/ch2_prog3.m :linenos: True :language: matlab ::: Output: ```none t y f(t) y_ex Abs_error 0.00 1.0000000 0.0000000 1.0000000 0.00000000 0.10 1.0048374 0.0951626 1.0048374 0.00000001 0.20 1.0187308 0.1812692 1.0187308 0.00000006 0.30 1.0408184 0.2591816 1.0408182 0.00000018 0.40 1.0703202 0.3296798 1.0703200 0.00000013 0.50 1.1065307 0.3934693 1.1065307 0.00000007 0.60 1.1488116 0.4511884 1.1488116 0.00000000 0.70 1.1965852 0.5034148 1.1965853 0.00000006 0.80 1.2493289 0.5506711 1.2493290 0.00000011 0.90 1.3065696 0.5934304 1.3065697 0.00000009 1.00 1.3678794 0.6321206 1.3678794 0.00000008 ``` ```{figure} /images/fig-ch3program3_result.svg --- width: 600px name: fig-ch3program3_result --- ```