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Waiting for other notebooks to finish Pkg operations... === Resolving... === ┌ Warning: Pkg operation failed. Fixing stdlib dependencies and trying again... └ @ GracefulPkg ~/.julia/packages/GracefulPkg/GQ6My/src/apply strategies.jl:96 Resolving... ===  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_diejgcjjxr/Project.toml`  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_diejgcjjxr/Manifest.toml` Instantiating... === Precompiling... ===Plots Waiting for other notebooks to finish Pkg operations... === Resolving... === ┌ Warning: Pkg operation failed. Fixing stdlib dependencies and trying again... └ @ GracefulPkg ~/.julia/packages/GracefulPkg/GQ6My/src/apply strategies.jl:96 Resolving... ===  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_diejgcjjxr/Project.toml`  No Changes to `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_diejgcjjxr/Manifest.toml` Instantiating... === Precompiling... ===enabledìinstantiated÷restart_recommended_msgrestart_required_msginstall_time_nsyobusy_packagescell_inputs*$f6c91ba4-661e-4b6c-8722-de27e36f0e24cell_id$f6c91ba4-661e-4b6c-8722-de27e36f0e24codeexp(0.2)metadatashow_logsèdisabled®skip_as_script«code_folded$1fa787ef-324f-4cca-b5b7-f270f84a974bcell_id$1fa787ef-324f-4cca-b5b7-f270f84a974bcode1 + 1metadatashow_logsèdisabled®skip_as_script«code_folded$6fbf3bb2-61d2-4f55-a0f3-8a05ef6095e7cell_id$6fbf3bb2-61d2-4f55-a0f3-8a05ef6095e7codemd""" ## Numerical implementation Let us begin by __storing__ relevant information into __variables__. As a rough approximation, one can think of a computer as doing one of three things: - storing data - computing with data - displaying data """metadatashow_logsèdisabled®skip_as_script«code_folded$f2fcef8a-e753-488a-a09a-be6c473755e8cell_id$f2fcef8a-e753-488a-a09a-be6c473755e8codef_exact = ℯ^ymetadatashow_logsèdisabled®skip_as_script«code_folded$8837fb56-f8a6-4152-9e50-bd3f465bbb02cell_id$8837fb56-f8a6-4152-9e50-bd3f465bbb02codecalc_f_residual(0.2, 10)metadatashow_logsèdisabled®skip_as_script«code_folded$7cac38d1-0524-4e66-8bc5-87bc5708ec4dcell_id$7cac38d1-0524-4e66-8bc5-87bc5708ec4dcodeM = 3metadatashow_logsèdisabled®skip_as_script«code_folded$3687516a-5a63-45c9-a2c9-415c706f45bbcell_id$3687516a-5a63-45c9-a2c9-415c706f45bbcodegmd""" ### Example Supposed we are tasked with calculating $$\tilde{f} (0.2 \, ; y_0 = 0, M = 3)$$. """metadatashow_logsèdisabled®skip_as_script«code_folded$9cb651da-8e08-479a-81e8-24de523c145ecell_id$9cb651da-8e08-479a-81e8-24de523c145ecodeℯ^0.2metadatashow_logsèdisabled®skip_as_script«code_folded$6f72cf27-57f7-490d-b076-1fc368577fe2cell_id$6f72cf27-57f7-490d-b076-1fc368577fe2codeπ * 2metadatashow_logsèdisabled®skip_as_script«code_folded$56e12061-2cbf-4c38-914e-7c75ffb9f33ecell_id$56e12061-2cbf-4c38-914e-7c75ffb9f33ecodeFmd""" !!! question "How do we plot the Taylor series approximation in a more useful way?" Notice that the above plot is not particularly helpful, because we cannot easily observe that our estimate of $$e^{0.2}$$ is better for larger `M`. __*In the first problem set, you will find better ways to plot the error.*__ """metadatashow_logsèdisabled®skip_as_script«code_folded$1536fbc1-0c35-4cdb-b663-9ae7fe0771f0cell_id$1536fbc1-0c35-4cdb-b663-9ae7fe0771f0code,abs(ℯ^0.2 - (1 + 0.2 + 0.2^2/2 + 0.2^3/6))metadatashow_logsèdisabled®skip_as_script«code_folded$fc8dd886-86e3-4c2d-9f3e-490b04c8e369cell_id$fc8dd886-86e3-4c2d-9f3e-490b04c8e369code3md""" What is the truncation error that arises? """metadatashow_logsèdisabled®skip_as_script«code_folded$3f635ff2-481c-4ac7-b3b7-14a28b3042c6cell_id$3f635ff2-481c-4ac7-b3b7-14a28b3042c6code1 + 0.2 + 0.2^2/2 + 0.2^3/6metadatashow_logsèdisabled®skip_as_script«code_folded$444ecadf-c50d-4fb9-bf9c-ca3c90abe0f7cell_id$444ecadf-c50d-4fb9-bf9c-ca3c90abe0f7code"residual = abs(f_exact - f_approx)metadatashow_logsèdisabled®skip_as_script«code_folded$268ee3a6-ece9-4c3f-a5bd-c1ea2835ce21cell_id$268ee3a6-ece9-4c3f-a5bd-c1ea2835ce21code$abs(ℯ^0.2 - calc_f_approx(0.2, 3))metadatashow_logsèdisabled®skip_as_script«code_folded$133c3077-80f3-4515-a7e5-dfc3d56eb12fcell_id$133c3077-80f3-4515-a7e5-dfc3d56eb12fcodeplot_f_approx(0.2, 6)metadatashow_logsèdisabled®skip_as_script«code_folded$3839ea7d-2de4-48dd-9002-767afa5c0b61cell_id$3839ea7d-2de4-48dd-9002-767afa5c0b61codeusing Plotsmetadatashow_logsèdisabled®skip_as_script«code_folded$a1735f28-8cd3-4784-9686-b567e28b857acell_id$a1735f28-8cd3-4784-9686-b567e28b857acodecalc_f_approx(0.2, 3)metadatashow_logsèdisabled®skip_as_script«code_folded$ca8ff74a-e249-4587-9d52-a7434b59e688cell_id$ca8ff74a-e249-4587-9d52-a7434b59e688codemd""" # Introduction > __Welcome to *CHE 348: Numerical Methods in Chemical Engineering* in Fall 2024!__ The significant increase in available compute over the last several decades has led to a transformation in the methods and techniques for solving science and engineering problems. To use new tools effectively, it is crucial that we understand their *theoretical underpinnings* and *limitations,* as well as *practical details of their implementation.* __CHE 348__ is a one-semester advanced undergraduate course providing students with a fundamental understanding of *common numerical methods* spanning * chemical engineering, * computational science, and * applied mathematics. In problems studied, __the underlying physics is emphasized throughout,__ with principles from __*non-dimensionalization*__ frequently employed. While the ideas presented are general (and thus not tied to any specific programming language), the entirety of the course will be carried out in [Julia](https://julialang.org/)—a powerful, easy-to-use language designed for scientists and engineers. ## Administrative items Lectures will be uploaded to the [course website](https://sahu-lab.github.io/teaching/che348/), and problem sets will be submitted via [canvas](https://utexas.instructure.com/courses/1385450) In the discussion section yesterday (26 Aug.), you should have [downloaded](https://julialang.org/downloads/) Julia, installed it along with the [Pluto](https://plutojl.org/) package. [Video instructions](https://computationalthinking.mit.edu/Fall23/installation/) are provided to aid you in this. ## Starting with Julia Launch the __REPL__ and verify that Julia is installed appropriately. You should be able to do the following on your machine: """metadatashow_logsèdisabled®skip_as_script«code_folded$0a8968f6-323e-4e17-b1aa-0560f78d7cdecell_id$0a8968f6-323e-4e17-b1aa-0560f78d7cdecodecalc_f_residual(0.2, 15)metadatashow_logsèdisabled®skip_as_script«code_folded$08e0e572-edb6-4524-97fe-41efb437cd8ccell_id$08e0e572-edb6-4524-97fe-41efb437cd8ccode2^-52metadatashow_logsèdisabled®skip_as_script«code_folded$b86837f1-2301-4ffb-92f9-890e0bd4e69fcell_id$b86837f1-2301-4ffb-92f9-890e0bd4e69fcodemd""" Analytically, we denote the truncation error—sometimes called the [__*residual*__](https://en.wikipedia.org/wiki/Residual_(numerical_analysis))—as $$r(y \, ; y_0, M)$$, which is defined to be ```math \begin{aligned} r(y \, ; y_0, M) \, :=& \ \big\lvert f(y) \, - \, \tilde{f} (y \, ; y_0, M) \big\rvert \\[4pt] \, =& \, \sum_{n = M + 1}^\infty \dfrac{1}{n!} \, \dfrac{\mathrm{d}^n f(y)}{\mathrm{d} y^n} \bigg\rvert_{y = y_0} \big(y - y_0 \big)^{\! n} ~. \end{aligned} ``` Notice that if the $$n^{\text{th}}$$ derivative of the function $$f$$ is "of order unity," i.e. it is relatively constant for all $$n$$, then as $$n$$ increases the $$n!$$ in the denominator will become larger than the $$(y - y_0)^n$$ in the numerator. When using Taylor series, we are generally concerned with how large the residual is for a given choice of $$y_0$$ and $$M$$. > __*You will investigate this dependence in Problem Set 1*__ Note that you could, in principle, calculate all these errors—for many different choices of $$y_0$$ and $$M$$— one by one. However, these sorts of repetitive tasks are much more conveniently done with a computer! Let us begin to see how to do so. """metadatashow_logsèdisabled®skip_as_script«code_folded$6083bfe2-79e6-4f2d-b511-bb904d4d6838cell_id$6083bfe2-79e6-4f2d-b511-bb904d4d6838codecalc_f_residual(0.2, 12)metadatashow_logsèdisabled®skip_as_script«code_folded$f43442e6-8ca3-4def-b8d0-ac3a325ddd3bcell_id$f43442e6-8ca3-4def-b8d0-ac3a325ddd3bcode#md""" !!! question "Why does our error stop decreasing?" Notice that after a certain value of `M` (11 in this case), the residual does not continue to decrease—even though we would naively expect it to, as we are including more terms in the Taylor series. How are these two observations consistent? > __Only a finite amount of memory__ is allocated for any variable, which means that *__very__ small changes to a variable will not be recorded!* > This is the concept of [machine precision](https://en.wikipedia.org/wiki/Machine_epsilon), which will be talked about in discussion section #3. > As a preview, compare the "error floor" above with the value of $$2^{-52}$$ below. > > *Note: this agreement is specific to how your computer stores numbers; in my case I have a 64-bit machine.* """metadatashow_logsèdisabled®skip_as_script«code_folded$12084602-3a09-47f3-864b-55cbac042795cell_id$12084602-3a09-47f3-864b-55cbac042795codeRmd""" ## References The following are some useful references, in case you want to learn more. This list is not exhaustive, as there are many great resources! - [3Blue1Brown](https://youtu.be/3d6DsjIBzJ4) - [Wikipedia](https://en.wikipedia.org/wiki/Taylor_series) - [Wolfram MathWorld](https://mathworld.wolfram.com/TaylorSeries.html) """metadatashow_logsèdisabled®skip_as_script«code_folded$d0225810-052b-480b-895c-16732badbf84cell_id$d0225810-052b-480b-895c-16732badbf84codey = 0.2metadatashow_logsèdisabled®skip_as_script«code_folded$40f12b03-98f0-4b13-b240-141f17146037cell_id$40f12b03-98f0-4b13-b240-141f17146037codecalc_f_residual(0.2, 11)metadatashow_logsèdisabled®skip_as_script«code_folded$b4669b9d-0846-49f6-a7eb-3f8ca31a25dfcell_id$b4669b9d-0846-49f6-a7eb-3f8ca31a25dfcodeDmd""" How does this compare to the actual value of the function? """metadatashow_logsèdisabled®skip_as_script«code_folded$363e3ad3-c1ce-4a6c-8113-96e170bfca7acell_id$363e3ad3-c1ce-4a6c-8113-96e170bfca7acode f_approx = 1 + y + y^2/2 + y^3/6metadatashow_logsèdisabled®skip_as_script«code_folded$1756de61-190e-4926-8c81-d5b7e9d6de10cell_id$1756de61-190e-4926-8c81-d5b7e9d6de10codecalc_f_residual(0.2, 9)metadatashow_logsèdisabled®skip_as_script«code_folded$8a7e706c-1934-42f8-8a7b-9f2b102ce2eccell_id$8a7e706c-1934-42f8-8a7b-9f2b102ce2eccodemd""" # Taylor series ## Review The [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) is often said to be the most important concept from calculus. From [Wikipedia](https://en.wikipedia.org/wiki/Taylor_series): > The Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Consider a function $$f(y)$$, where $$y \in \mathbb{C}$$ and $$f \in \mathbb{C}$$, for which one could write ```math \begin{aligned} f(y) \, &= \, f(y_0) \, + \, f'(y_0) \, (y - y_0) \, + \, \dfrac{1}{2!} f''(y_0) \, (y - y_0)^2 \, + \, \ldots \\[5pt] \, &= \, \sum_{n = 0}^\infty \dfrac{1}{n!} \, \dfrac{\mathrm{d}^n f(y)}{\mathrm{d} y^n} \bigg\rvert_{y = y_0} \big(y - y_0 \big)^{\! n} ~. \end{aligned} ``` The Taylor series yields many of the relations you have seen in the past. For example, ```math \mathrm{e}^y \, = \, 1 \, + \, y \, + \, \dfrac{y^2}{2!} \, + \, \dfrac{y^3}{3!} \, + \, \ldots ``` !!! question "Practice Question" What value is the above series expanded about? """metadatashow_logsèdisabled®skip_as_script«code_folded$395b20fe-7ddf-4b83-8e55-ed1a06714b49cell_id$395b20fe-7ddf-4b83-8e55-ed1a06714b49codefor n = 0:M println(n) endmetadatashow_logsèdisabled®skip_as_script«code_folded$3078d320-678d-4bb6-9742-4f3720bca392cell_id$3078d320-678d-4bb6-9742-4f3720bca392codeLfunction calc_f_residual(y, M) return abs(ℯ^y - calc_f_approx(y, M)); endmetadatashow_logsèdisabled®skip_as_script«code_folded$8a4e56fd-e98c-4022-bf9a-e76c3e64fda6cell_id$8a4e56fd-e98c-4022-bf9a-e76c3e64fda6codeنmd""" ### Initializing packages *When running this notebook for the first time, this could take up to 10 minutes. Hang in there!* """metadatashow_logsèdisabled®skip_as_script«code_folded$f2646139-cd94-425f-b517-4e1208fbdb71cell_id$f2646139-cd94-425f-b517-4e1208fbdb71codemd""" --- --- """metadatashow_logsèdisabled®skip_as_script«code_folded$16fd3897-0b76-47a8-8253-6e267cbd9960cell_id$16fd3897-0b76-47a8-8253-6e267cbd9960codetmd""" !!! info "Objective" Use a `for` loop to calculate `f_approx`, which works for different values of `M` """metadatashow_logsèdisabled®skip_as_script«code_folded$17da61cf-ce3f-4d92-8537-81eaf35a927acell_id$17da61cf-ce3f-4d92-8537-81eaf35a927acodecalc_f_residual(0.2, 20)metadatashow_logsèdisabled®skip_as_script«code_folded$4786c577-85ea-44f6-8a93-5c9e6233eb70cell_id$4786c577-85ea-44f6-8a93-5c9e6233eb70codeXfunction plot_f_approx(y, M_max) M_array = 1:M_max; f_approx_array = calc_f_approx.(y, M_array); Plots.plot( M_array, # x values f_approx_array, # y values title = "Taylor series approximation", # plot title label = "f_approx(y, M)", # legend xlabel = "M", # x-axis label ylabel = "f_approx", # y-axis label line=(2, :darkorange3), # line width and color marker=(:circle, 5, :darkorange3), # marker shape, size, and color ) endmetadatashow_logsèdisabled®skip_as_script«code_folded$6c753149-f441-48a8-b947-3d23f1cd30aecell_id$6c753149-f441-48a8-b947-3d23f1cd30aecodeٯmd""" !!! info "Objective" Plot the approximate calculation of $$e^{0.2}$$, based on the Taylor series expansion, as a function of `M`—where `M` goes from `1` to `6` """metadatashow_logsèdisabled®skip_as_script«code_folded$f35c69f2-74c5-4001-a3f6-4d10e96ad4dfcell_id$f35c69f2-74c5-4001-a3f6-4d10e96ad4dfcodemd""" ## Numerical approximation In a computer, one cannot sum an infinite number of terms. A __[truncation error](https://en.wikipedia.org/wiki/Truncation_error)__ arises because only a finite number of terms are included. Let us __*define*__ the function $$\tilde{f} (y \, ; y_0, M)$$ as the truncated polynomial series of order $$M$$, written as ```math \tilde{f} (y \,; y_0, M) \, := \, \sum_{n = 0}^M \dfrac{1}{n!} \, \dfrac{\mathrm{d}^n f(y)}{\mathrm{d} y^n} \bigg\rvert_{y = y_0} \big(y - y_0 \big)^{\! n} ~. ``` !!! question "Practice Question" How many terms are in $$\tilde{f} (y \, ; y_0, M)$$? !!! question "Practice Question" What is the highest-order polynomial term in $$\tilde{f} (y \, ; y_0, M)$$? Let us now use the exponential function as an example. To this end, we *choose* ```math f(y) \, = \, \mathrm{e}^y ~, ``` for which ```math \tilde{f} (y \, ; y_0 = 0, M) \, := \, 1 \, + \, y \, + \, \dfrac{y^2}{2!} \, + \, \dfrac{y^3}{3!} \, + \, \ldots \, + \, \dfrac{y^M}{M!} ~. ``` --- --- """metadatashow_logsèdisabled®skip_as_script«code_folded$98d6e55e-a63b-411c-b7cb-e5dd15e686f9cell_id$98d6e55e-a63b-411c-b7cb-e5dd15e686f9codeفfunction calc_f_approx(y, M) f_approx = 0.0; for n = 0:M f_approx = f_approx + y^n / factorial(n); end return f_approx; endmetadatashow_logsèdisabled®skip_as_script«code_folded$a9b6fcd4-9a8e-4d0d-bd13-e0dc318adbd3cell_id$a9b6fcd4-9a8e-4d0d-bd13-e0dc318adbd3codeMmd""" So far, our calculation of `f_approx` does not use the value of `M` that we set. There is an easy way for us to tell the computer to __span__ a range of values! ```julia # the following code executes a "for loop" # notice that lines starting with "#" are comments (they are not executed) for n = 0:M println(n) end ``` """metadatashow_logsèdisabled®skip_as_script«code_foldedënotebook_id$65e0a2ce-9d9e-11f1-04b4-af365d5badbfbondscell_results*$f6c91ba4-661e-4b6c-8722-de27e36f0e24queued¤logsrunning¦outputbody1.2214027581601699persist_js_state¤mimetext/plainlast_run_timestampAڢ- zkhas_pluto_hook_features¬rootassigneecell_id$f6c91ba4-661e-4b6c-8722-de27e36f0e24depends_on_disabled_cells§runtime&published_object_keysdepends_on_skipped_cells§errored$1fa787ef-324f-4cca-b5b7-f270f84a974bqueued¤logsrunning¦outputbody2persist_js_state¤mimetext/plainlast_run_timestampAڢ- jhas_pluto_hook_features¬rootassigneecell_id$1fa787ef-324f-4cca-b5b7-f270f84a974bdepends_on_disabled_cells§runtime)published_object_keysdepends_on_skipped_cells§errored$6fbf3bb2-61d2-4f55-a0f3-8a05ef6095e7queued¤logsrunning¦outputbody

Numerical implementation

Let us begin by storing relevant information into variables. As a rough approximation, one can think of a computer as doing one of three things:

persist_js_state¤mimetext/htmllast_run_timestampAڢ-has_pluto_hook_features¬rootassigneecell_id$6fbf3bb2-61d2-4f55-a0f3-8a05ef6095e7depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$f2fcef8a-e753-488a-a09a-be6c473755e8queued¤logsrunning¦outputbody1.2214027581601699persist_js_state¤mimetext/plainlast_run_timestampAڢ- Pvhas_pluto_hook_features¬rootassigneef_exactcell_id$f2fcef8a-e753-488a-a09a-be6c473755e8depends_on_disabled_cells§runtime;published_object_keysdepends_on_skipped_cells§errored$8837fb56-f8a6-4152-9e50-bd3f465bbb02queued¤logsrunning¦outputbody6.661338147750939e-16persist_js_state¤mimetext/plainlast_run_timestampAڢ- ŷhas_pluto_hook_features¬rootassigneecell_id$8837fb56-f8a6-4152-9e50-bd3f465bbb02depends_on_disabled_cells§runtime/published_object_keysdepends_on_skipped_cells§errored$7cac38d1-0524-4e66-8bc5-87bc5708ec4dqueued¤logsrunning¦outputbody3persist_js_state¤mimetext/plainlast_run_timestampAڢ- ˼has_pluto_hook_features¬rootassigneeMcell_id$7cac38d1-0524-4e66-8bc5-87bc5708ec4ddepends_on_disabled_cells§runtime*published_object_keysdepends_on_skipped_cells§errored$3687516a-5a63-45c9-a2c9-415c706f45bbqueued¤logsrunning¦outputbody

Example

Supposed we are tasked with calculating $\tilde{f} (0.2 \, ; y_0 = 0, M = 3)$.

persist_js_state¤mimetext/htmllast_run_timestampAڢ-phas_pluto_hook_features¬rootassigneecell_id$3687516a-5a63-45c9-a2c9-415c706f45bbdepends_on_disabled_cells§runtimeupublished_object_keysdepends_on_skipped_cells§errored$9cb651da-8e08-479a-81e8-24de523c145equeued¤logsrunning¦outputbody1.2214027581601699persist_js_state¤mimetext/plainlast_run_timestampAڢ- zhas_pluto_hook_features¬rootassigneecell_id$9cb651da-8e08-479a-81e8-24de523c145edepends_on_disabled_cells§runtime-=published_object_keysdepends_on_skipped_cells§errored$6f72cf27-57f7-490d-b076-1fc368577fe2queued¤logsrunning¦outputbody6.283185307179586persist_js_state¤mimetext/plainlast_run_timestampAڢ- scshas_pluto_hook_features¬rootassigneecell_id$6f72cf27-57f7-490d-b076-1fc368577fe2depends_on_disabled_cells§runtime,%published_object_keysdepends_on_skipped_cells§errored$56e12061-2cbf-4c38-914e-7c75ffb9f33equeued¤logsrunning¦outputbody

How do we plot the Taylor series approximation in a more useful way?

Notice that the above plot is not particularly helpful, because we cannot easily observe that our estimate of $e^{0.2}$ is better for larger M. In the first problem set, you will find better ways to plot the error.

persist_js_state¤mimetext/htmllast_run_timestampAڢ-;`has_pluto_hook_features¬rootassigneecell_id$56e12061-2cbf-4c38-914e-7c75ffb9f33edepends_on_disabled_cells§runtimeFpublished_object_keysdepends_on_skipped_cells§errored$1536fbc1-0c35-4cdb-b663-9ae7fe0771f0queued¤logsrunning¦outputbody6.942482683647277e-5persist_js_state¤mimetext/plainlast_run_timestampAڢ- {Phas_pluto_hook_features¬rootassigneecell_id$1536fbc1-0c35-4cdb-b663-9ae7fe0771f0depends_on_disabled_cells§runtimeYpublished_object_keysdepends_on_skipped_cells§errored$fc8dd886-86e3-4c2d-9f3e-490b04c8e369queued¤logsrunning¦outputbodyM

What is the truncation error that arises?

persist_js_state¤mimetext/htmllast_run_timestampAڢ-Bhas_pluto_hook_features¬rootassigneecell_id$fc8dd886-86e3-4c2d-9f3e-490b04c8e369depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$3f635ff2-481c-4ac7-b3b7-14a28b3042c6queued¤logsrunning¦outputbody1.2213333333333334persist_js_state¤mimetext/plainlast_run_timestampAڢ- yfhas_pluto_hook_features¬rootassigneecell_id$3f635ff2-481c-4ac7-b3b7-14a28b3042c6depends_on_disabled_cells§runtimedɵpublished_object_keysdepends_on_skipped_cells§errored$444ecadf-c50d-4fb9-bf9c-ca3c90abe0f7queued¤logsrunning¦outputbody6.942482683647277e-5persist_js_state¤mimetext/plainlast_run_timestampAڢ- ,has_pluto_hook_features¬rootassigneeresidualcell_id$444ecadf-c50d-4fb9-bf9c-ca3c90abe0f7depends_on_disabled_cells§runtime;published_object_keysdepends_on_skipped_cells§errored$268ee3a6-ece9-4c3f-a5bd-c1ea2835ce21queued¤logsrunning¦outputbody6.942482683647277e-5persist_js_state¤mimetext/plainlast_run_timestampAڢ- rhas_pluto_hook_features¬rootassigneecell_id$268ee3a6-ece9-4c3f-a5bd-c1ea2835ce21depends_on_disabled_cells§runtimeCypublished_object_keysdepends_on_skipped_cells§errored$133c3077-80f3-4515-a7e5-dfc3d56eb12fqueued¤logsrunning¦outputbodyȿ  persist_js_state¤mimeimage/svg+xmllast_run_timestampAڢ- {has_pluto_hook_features¬rootassigneecell_id$133c3077-80f3-4515-a7e5-dfc3d56eb12fdepends_on_disabled_cells§runtimeZ{published_object_keysdepends_on_skipped_cells§errored$3839ea7d-2de4-48dd-9002-767afa5c0b61queued¤logsrunning¦outputbodypersist_js_state¤mimetext/plainlast_run_timestampAڢ- Lhas_pluto_hook_features¬rootassigneecell_id$3839ea7d-2de4-48dd-9002-767afa5c0b61depends_on_disabled_cells§runtimeLappublished_object_keysdepends_on_skipped_cells§errored$a1735f28-8cd3-4784-9686-b567e28b857aqueued¤logsrunning¦outputbody1.2213333333333334persist_js_state¤mimetext/plainlast_run_timestampAڢ- Qhas_pluto_hook_features¬rootassigneecell_id$a1735f28-8cd3-4784-9686-b567e28b857adepends_on_disabled_cells§runtime/published_object_keysdepends_on_skipped_cells§errored$ca8ff74a-e249-4587-9d52-a7434b59e688queued¤logsrunning¦outputbody

Introduction

Welcome to CHE 348: Numerical Methods in Chemical Engineering in Fall 2024!

The significant increase in available compute over the last several decades has led to a transformation in the methods and techniques for solving science and engineering problems. To use new tools effectively, it is crucial that we understand their theoretical underpinnings and limitations, as well as practical details of their implementation.

CHE 348 is a one-semester advanced undergraduate course providing students with a fundamental understanding of common numerical methods spanning

In problems studied, the underlying physics is emphasized throughout, with principles from non-dimensionalization frequently employed.

While the ideas presented are general (and thus not tied to any specific programming language), the entirety of the course will be carried out in Julia—a powerful, easy-to-use language designed for scientists and engineers.

Administrative items

Lectures will be uploaded to the course website, and problem sets will be submitted via canvas

In the discussion section yesterday (26 Aug.), you should have downloaded Julia, installed it along with the Pluto package. Video instructions are provided to aid you in this.

Starting with Julia

Launch the REPL and verify that Julia is installed appropriately. You should be able to do the following on your machine:

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Analytically, we denote the truncation error—sometimes called the residual—as $r(y \, ; y_0, M)$, which is defined to be

$$\begin{aligned} r(y \, ; y_0, M) \, :=& \ \big\lvert f(y) \, - \, \tilde{f} (y \, ; y_0, M) \big\rvert \\[4pt] \, =& \, \sum_{n = M + 1}^\infty \dfrac{1}{n!} \, \dfrac{\mathrm{d}^n f(y)}{\mathrm{d} y^n} \bigg\rvert_{y = y_0} \big(y - y_0 \big)^{\! n} ~. \end{aligned}$$

Notice that if the $n^{\text{th}}$ derivative of the function $f$ is "of order unity," i.e. it is relatively constant for all $n$, then as $n$ increases the $n!$ in the denominator will become larger than the $(y - y_0)^n$ in the numerator. When using Taylor series, we are generally concerned with how large the residual is for a given choice of $y_0$ and $M$.

You will investigate this dependence in Problem Set 1

Note that you could, in principle, calculate all these errors—for many different choices of $y_0$ and $M$— one by one. However, these sorts of repetitive tasks are much more conveniently done with a computer! Let us begin to see how to do so.

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Why does our error stop decreasing?

Notice that after a certain value of M (11 in this case), the residual does not continue to decrease—even though we would naively expect it to, as we are including more terms in the Taylor series. How are these two observations consistent?

Only a finite amount of memory is allocated for any variable, which means that very small changes to a variable will not be recorded! This is the concept of machine precision, which will be talked about in discussion section #3. As a preview, compare the "error floor" above with the value of $2^{-52}$ below.

Note: this agreement is specific to how your computer stores numbers; in my case I have a 64-bit machine.

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References

The following are some useful references, in case you want to learn more. This list is not exhaustive, as there are many great resources!

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How does this compare to the actual value of the function?

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Taylor series

Review

The Taylor series is often said to be the most important concept from calculus. From Wikipedia:

The Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.

Consider a function $f(y)$, where $y \in \mathbb{C}$ and $f \in \mathbb{C}$, for which one could write

$$\begin{aligned} f(y) \, &= \, f(y_0) \, + \, f'(y_0) \, (y - y_0) \, + \, \dfrac{1}{2!} f''(y_0) \, (y - y_0)^2 \, + \, \ldots \\[5pt] \, &= \, \sum_{n = 0}^\infty \dfrac{1}{n!} \, \dfrac{\mathrm{d}^n f(y)}{\mathrm{d} y^n} \bigg\rvert_{y = y_0} \big(y - y_0 \big)^{\! n} ~. \end{aligned}$$

The Taylor series yields many of the relations you have seen in the past. For example,

$$\mathrm{e}^y \, = \, 1 \, + \, y \, + \, \dfrac{y^2}{2!} \, + \, \dfrac{y^3}{3!} \, + \, \ldots$$

Practice Question

What value is the above series expanded about?

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Initializing packages

When running this notebook for the first time, this could take up to 10 minutes. Hang in there!

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Objective

Use a for loop to calculate f_approx, which works for different values of M

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Objective

Plot the approximate calculation of $e^{0.2}$, based on the Taylor series expansion, as a function of M—where M goes from 1 to 6

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Numerical approximation

In a computer, one cannot sum an infinite number of terms. A truncation error arises because only a finite number of terms are included.

Let us define the function $\tilde{f} (y \, ; y_0, M)$ as the truncated polynomial series of order $M$, written as

$$\tilde{f} (y \,; y_0, M) \, := \, \sum_{n = 0}^M \dfrac{1}{n!} \, \dfrac{\mathrm{d}^n f(y)}{\mathrm{d} y^n} \bigg\rvert_{y = y_0} \big(y - y_0 \big)^{\! n} ~.$$

Practice Question

How many terms are in $\tilde{f} (y \, ; y_0, M)$?

Practice Question

What is the highest-order polynomial term in $\tilde{f} (y \, ; y_0, M)$?

Let us now use the exponential function as an example. To this end, we choose

$$f(y) \, = \, \mathrm{e}^y ~,$$

for which

$$\tilde{f} (y \, ; y_0 = 0, M) \, := \, 1 \, + \, y \, + \, \dfrac{y^2}{2!} \, + \, \dfrac{y^3}{3!} \, + \, \ldots \, + \, \dfrac{y^M}{M!} ~.$$

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So far, our calculation of f_approx does not use the value of M that we set. There is an easy way for us to tell the computer to span a range of values!

# the following code executes a "for loop"
# notice that lines starting with "#" are comments (they are not executed)

for n = 0:M
    println(n)
end
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