import Pkg
Pkg.add("Plots")
using Plots
# Parameters
L = 1.0 # Length of domain
Nx = 100 # Number of spatial grid points
dx = L/(Nx-1) # Spatial step
D = 0.01 # Diffusion coefficient
dt = 0.0005 # Time step
Nt = 400 # Number of time steps
x = LinRange(0, L, Nx)
# Finite difference Laplacian (second derivative)
function laplacian(u, dx)
du = similar(u)
du[1] = 0.0 # Neumann BC: du/dx = 0 at boundaries
du[end] = 0.0
for i in 2:(length(u)-1)
du[i] = (u[i-1] - 2u[i] + u[i+1]) / dx^2
end
return du
end
# Time stepping (Forward Euler)
function diffuse(u0, D, dx, dt, Nt)
u = copy(u0)
sol = [copy(u)]
for n in 1:Nt
u .= u .+ D*dt .* laplacian(u, dx)
push!(sol, copy(u))
end
return sol
end
# Different initial conditions
u0_gaussian = exp.(-(x .- 0.5).^2 ./ 0.01)
u0_step = x .< 0.5
u0_sine = sin.(π*x)
# Solve
sol_gaussian = diffuse(u0_gaussian, D, dx, dt, Nt)
#sol_step = diffuse(u0_step, D, dx, dt, Nt)
sol_sine = diffuse(u0_sine, D, dx, dt, Nt)
# Plot snapshots
function plot_solution(sol, x; title_str="Diffusion")
times = [1, round(Int,Nt/4), round(Int,Nt/2), Nt]
plot(x, sol[1], label="t=0")
for t in times[2:end]
plot!(x, sol[t], label="t=$(round(t*dt, digits=2))")
end
xlabel!("x")
ylabel!("u(x,t)")
title!(title_str)
end
plot1 = plot_solution(sol_gaussian, x, title_str="Gaussian Initial Condition")
#plot2 = plot_solution(sol_step, x, title_str="Step Initial Condition")
plot3 = plot_solution(sol_sine, x, title_str="Sine Initial Condition")
plot(plot1, plot3, layout=(3,1), size=(700,900))Resolving package versions... Installed x265_jll ───────────────────── v4.1.0+0 Installed libdecor_jll ───────────────── v0.2.2+0 Installed GR_jll ─────────────────────── v0.73.26+0 Installed libfdk_aac_jll ─────────────── v2.0.4+0 Installed Libmount_jll ───────────────── v2.42.0+0 Installed LERC_jll ───────────────────── v4.1.0+0 Installed JpegTurbo_jll ──────────────── v3.1.5+0 Installed LoggingExtras ──────────────── v1.2.0 Installed Opus_jll ───────────────────── v1.6.1+0 Installed ConcurrentUtilities ────────── v2.5.1 Installed Xorg_xkbcomp_jll ───────────── v1.4.7+0 Installed Grisu ──────────────────────── v1.0.2 Installed GR ─────────────────────────── v0.73.26 Installed Measures ───────────────────── v0.3.3 Installed Contour ────────────────────── v0.6.3 Installed Xorg_xcb_util_wm_jll ───────── v0.4.2+0 Installed Xorg_xcb_util_image_jll ────── v0.4.1+0 Installed RecipesPipeline ────────────── v0.6.12 Installed PlotUtils ──────────────────── v1.4.4 Installed DelimitedFiles ─────────────── v1.9.1 Installed OpenSSL ────────────────────── v1.6.1 Installed Xorg_libSM_jll ─────────────── v1.2.6+0 Installed Cairo_jll ──────────────────── v1.18.7+0 Installed HTTP ───────────────────────── v1.11.0 Installed Xorg_xcb_util_jll ──────────── v0.4.1+0 Installed libva_jll ──────────────────── v2.23.0+0 Installed Fontconfig_jll ─────────────── v2.17.1+0 Installed Statistics ─────────────────── v1.11.1 Installed Xorg_libxkbfile_jll ────────── v1.2.0+0 Installed Xorg_libXinerama_jll ───────── v1.1.7+0 Installed Xorg_libXau_jll ────────────── v1.0.13+0 Installed EpollShim_jll ──────────────── v0.0.20230411+1 Installed Missings ───────────────────── v1.2.0 Installed IrrationalConstants ────────── v0.2.6 Installed FFMPEG ─────────────────────── v0.4.5 Installed Showoff ────────────────────── v1.0.3 Installed Pango_jll ──────────────────── v1.57.1+0 Installed Xorg_libpciaccess_jll ──────── v0.19.0+0 Installed Xorg_xcb_util_keysyms_jll ──── v0.4.1+0 Installed PtrArrays ──────────────────── v1.4.0 Installed xkbcommon_jll ──────────────── v1.13.0+0 Installed Bzip2_jll ──────────────────── v1.0.9+0 Installed SimpleBufferStream ─────────── v1.2.0 Installed HarfBuzz_jll ───────────────── v8.5.1+0 Installed XZ_jll ─────────────────────── v5.8.3+0 Installed PlotThemes ─────────────────── v3.3.0 Installed NaNMath ────────────────────── v1.1.4 Installed fzf_jll ────────────────────── v0.61.1+0 Installed FriBidi_jll ────────────────── v1.0.17+0 Installed UnicodeFun ─────────────────── v0.4.1 Installed TranscodingStreams ─────────── v0.11.3 Installed MbedTLS ────────────────────── v1.1.10 Installed GLFW_jll ───────────────────── v3.4.1+1 Installed x264_jll ───────────────────── v10164.0.1+0 Installed ColorSchemes ───────────────── v3.31.0 Installed FreeType2_jll ──────────────── v2.14.3+1 Installed JLFzf ──────────────────────── v0.1.11 Installed DataStructures ─────────────── v0.19.5 Installed StatsAPI ───────────────────── v1.8.0 Installed CodecZlib ──────────────────── v0.7.8 Installed Colors 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artifact Libtiff 1.5 MiB Installed artifact x264 1.8 MiB Installed artifact LERC 222.7 KiB Installed artifact libvorbis 290.9 KiB Installed artifact Pango 613.2 KiB Installed artifact Zstd 568.6 KiB Installed artifact FreeType2 601.1 KiB Installed artifact fzf 3.2 MiB Installed artifact HarfBuzz 1.4 MiB Installed artifact MbedTLS 1.9 MiB Installed artifact Fontconfig 203.1 KiB Installed artifact Opus 339.3 KiB Installed artifact libass 145.0 KiB Installed artifact GettextRuntime 312.5 KiB Installed artifact FriBidi 63.8 KiB Installed artifact Xorg_xkeyboard_config 545.0 KiB Installed artifact libpng 316.6 KiB Installed artifact Expat 145.9 KiB Installed artifact libaom 6.5 MiB Installed artifact LAME 315.8 KiB Installed artifact Vulkan_Loader 282.9 KiB Installed artifact LLVMOpenMP 340.9 KiB Installed artifact Qt6ShaderTools 1.8 MiB Installed artifact Cairo 764.9 KiB Installed artifact Glib 7.0 MiB Installed artifact x265 1.3 MiB Installed artifact FFMPEG 12.5 MiB Installed artifact 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+ libfdk_aac_jll v2.0.4+0 [36db933b] + libinput_jll v1.28.1+0 [b53b4c65] + libpng_jll v1.6.58+0 [9a156e7d] + libva_jll v2.23.0+0 [f27f6e37] + libvorbis_jll v1.3.8+0 [009596ad] + mtdev_jll v1.1.7+0 ⌅ [1270edf5] + x264_jll v10164.0.1+0 [dfaa095f] + x265_jll v4.1.0+0 [d8fb68d0] + xkbcommon_jll v1.13.0+0 [0dad84c5] + ArgTools v1.1.2 [56f22d72] + Artifacts v1.11.0 [2a0f44e3] + Base64 v1.11.0 [ade2ca70] + Dates v1.11.0 [f43a241f] + Downloads v1.7.0 [7b1f6079] + FileWatching v1.11.0 [b77e0a4c] + InteractiveUtils v1.11.0 [ac6e5ff7] + JuliaSyntaxHighlighting v1.12.0 [b27032c2] + LibCURL v0.6.4 [76f85450] + LibGit2 v1.11.0 [8f399da3] + Libdl v1.11.0 [37e2e46d] + LinearAlgebra v1.12.0 [56ddb016] + Logging v1.11.0 [d6f4376e] + Markdown v1.11.0 [a63ad114] + Mmap v1.11.0 [ca575930] + NetworkOptions v1.3.0 [44cfe95a] + Pkg v1.12.1 [de0858da] + Printf v1.11.0 [3fa0cd96] + REPL v1.11.0 [9a3f8284] + Random v1.11.0 [ea8e919c] + SHA v0.7.0 [9e88b42a] + Serialization v1.11.0 [6462fe0b] + Sockets v1.11.0 [2f01184e] + SparseArrays v1.12.0 [f489334b] + StyledStrings v1.11.0 [fa267f1f] + TOML v1.0.3 [a4e569a6] + Tar v1.10.0 [8dfed614] + Test v1.11.0 [cf7118a7] + UUIDs v1.11.0 [4ec0a83e] + Unicode v1.11.0 [e66e0078] + CompilerSupportLibraries_jll v1.3.0+1 [deac9b47] + LibCURL_jll v8.15.0+0 [e37daf67] + LibGit2_jll v1.9.0+0 [29816b5a] + LibSSH2_jll v1.11.3+1 [14a3606d] + MozillaCACerts_jll v2025.11.4 [4536629a] + OpenBLAS_jll v0.3.29+0 [05823500] + OpenLibm_jll v0.8.7+0 [458c3c95] + OpenSSL_jll v3.5.4+0 [efcefdf7] + PCRE2_jll v10.44.0+1 [bea87d4a] + SuiteSparse_jll v7.8.3+2 [83775a58] + Zlib_jll v1.3.1+2 [8e850b90] + libblastrampoline_jll v5.15.0+0 [8e850ede] + nghttp2_jll v1.64.0+1 [3f19e933] + p7zip_jll v17.7.0+0 Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m` Precompiling packages... 720.3 ms ✓ LaTeXStrings 941.5 ms ✓ TensorCore 1138.0 ms ✓ StatsAPI 1362.8 ms ✓ Contour 1796.4 ms ✓ Measures 2090.9 ms ✓ Statistics 2395.9 ms ✓ Format 3174.6 ms ✓ StableRNGs 3236.2 ms ✓ Requires 3729.2 ms ✓ DocStringExtensions 3776.9 ms ✓ Reexport 4098.0 ms ✓ OrderedCollections 4118.2 ms ✓ Grisu 3874.4 ms ✓ URIs 3733.5 ms ✓ SimpleBufferStream 4698.6 ms ✓ Unzip 3795.2 ms ✓ PtrArrays 5040.1 ms ✓ MacroTools 3998.8 ms ✓ TranscodingStreams 3543.0 ms ✓ DelimitedFiles 3409.5 ms ✓ DataAPI 2881.1 ms ✓ BitFlags 6115.4 ms ✓ IrrationalConstants 4810.0 ms ✓ NaNMath 3362.3 ms ✓ ConcurrentUtilities 2870.9 ms ✓ LoggingExtras 3012.6 ms ✓ ExceptionUnwrapping 3280.3 ms ✓ Graphite2_jll 2930.0 ms ✓ Libmount_jll 3868.2 ms ✓ UnicodeFun 3063.0 ms ✓ EpollShim_jll 3340.6 ms ✓ LLVMOpenMP_jll 3330.6 ms ✓ Bzip2_jll 3326.4 ms ✓ Xorg_libpciaccess_jll 3221.1 ms ✓ Xorg_libICE_jll 3157.6 ms ✓ Xorg_libXau_jll 3305.8 ms ✓ libpng_jll 3254.7 ms ✓ libfdk_aac_jll 3411.1 ms ✓ LAME_jll 3230.0 ms ✓ LERC_jll 3431.4 ms ✓ fzf_jll 3741.3 ms ✓ JpegTurbo_jll 3456.7 ms ✓ XZ_jll 3287.6 ms ✓ Ogg_jll 3278.8 ms ✓ mtdev_jll 3357.0 ms ✓ Xorg_libXdmcp_jll 3612.4 ms ✓ x265_jll 3497.3 ms ✓ x264_jll 3544.1 ms ✓ libaom_jll 3673.5 ms ✓ MbedTLS_jll 3643.3 ms ✓ Zstd_jll 3634.7 ms ✓ Expat_jll 3517.4 ms ✓ Opus_jll 3405.2 ms ✓ Xorg_xtrans_jll 3388.1 ms ✓ libevdev_jll 3413.6 ms ✓ Libffi_jll 3389.7 ms ✓ eudev_jll 3403.6 ms ✓ FriBidi_jll 3278.2 ms ✓ Libuuid_jll 3753.3 ms ✓ GettextRuntime_jll 4271.9 ms ✓ RecipesBase 3259.2 ms ✓ Showoff 3648.0 ms ✓ libdrm_jll 3538.8 ms ✓ Xorg_libSM_jll 6654.8 ms ✓ Statistics → SparseArraysExt 6257.6 ms ✓ AliasTables 7249.0 ms ✓ DataStructures 6773.9 ms ✓ CodecZlib 3965.2 ms ✓ Xorg_libxcb_jll 8396.8 ms ✓ FixedPointNumbers 7529.6 ms ✓ Missings 8085.6 ms ✓ OpenSSL 6926.8 ms ✓ JLFzf 3594.9 ms ✓ Dbus_jll 3509.2 ms ✓ Wayland_jll 3533.3 ms ✓ libinput_jll 10162.8 ms ✓ LogExpFunctions 10203.8 ms ✓ Pixman_jll 4168.4 ms ✓ Xorg_xcb_util_jll 10342.3 ms ✓ FreeType2_jll 4052.5 ms ✓ Xorg_libX11_jll 9843.8 ms ✓ libvorbis_jll 6193.4 ms ✓ SortingAlgorithms 2342.0 ms ✓ Xorg_xcb_util_image_jll 2560.7 ms ✓ Xorg_xcb_util_keysyms_jll 2772.7 ms ✓ Xorg_xcb_util_renderutil_jll 2985.0 ms ✓ Xorg_xcb_util_wm_jll 3270.1 ms ✓ Xorg_libXrender_jll 12904.1 ms ✓ Ghostscript_jll 3488.0 ms ✓ Xorg_libXext_jll 3709.8 ms ✓ Xorg_libXfixes_jll 3923.3 ms ✓ Xorg_libxkbfile_jll 2792.0 ms ✓ Xorg_xcb_util_cursor_jll 9090.8 ms ✓ ColorTypes 2594.3 ms ✓ Xorg_libXinerama_jll 2816.9 ms ✓ Xorg_libXrandr_jll 3059.7 ms ✓ Libglvnd_jll 3067.2 ms ✓ Xorg_libXcursor_jll 3277.1 ms ✓ Xorg_libXi_jll 3486.1 ms ✓ libva_jll 3762.8 ms ✓ Xorg_xkbcomp_jll 15168.3 ms ✓ MbedTLS 5342.1 ms ✓ ColorTypes → StyledStringsExt 16951.7 ms ✓ Glib_jll 4649.8 ms ✓ Xorg_xkeyboard_config_jll 14750.7 ms ✓ Fontconfig_jll 2203.0 ms ✓ xkbcommon_jll 8817.4 ms ✓ ColorVectorSpace 9674.8 ms ✓ Colors 22192.7 ms ✓ Libtiff_jll 2326.8 ms ✓ Vulkan_Loader_jll 17584.4 ms ✓ StatsBase 16041.7 ms ✓ Latexify 2256.4 ms ✓ Latexify → SparseArraysExt 6546.0 ms ✓ ColorSchemes 11619.8 ms ✓ Cairo_jll 10682.3 ms ✓ Qt6Base_jll 3666.0 ms ✓ HarfBuzz_jll 9803.4 ms ✓ PlotUtils 26165.8 ms ✓ HTTP 6087.7 ms ✓ libass_jll 6177.1 ms ✓ Pango_jll 4280.1 ms ✓ PlotThemes 9179.8 ms ✓ Qt6ShaderTools_jll 9218.1 ms ✓ Qt6Svg_jll 1345.3 ms ✓ libdecor_jll 6310.3 ms ✓ RecipesPipeline 1235.9 ms ✓ GLFW_jll 5555.8 ms ✓ Qt6Declarative_jll 7615.1 ms ✓ FFMPEG_jll 1003.6 ms ✓ Qt6Wayland_jll 3011.7 ms ✓ FFMPEG 4247.2 ms ✓ GR_jll 5129.9 ms ✓ GR 48595.5 ms ✓ Plots 135 dependencies successfully precompiled in 125 seconds. 45 already precompiled. Precompiling packages... 1510.5 ms ✓ QuartoNotebookWorkerJSONExt (serial) 1 dependency successfully precompiled in 2 seconds Precompiling packages... 3347.1 ms ✓ QuartoNotebookWorkerPlotsExt (serial) 1 dependency successfully precompiled in 3 seconds
