From 995f7ec22abf9b03e1e1051fcaf5aced67555fee Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Wed, 22 Apr 2026 11:51:11 -0700 Subject: [PATCH 01/15] Adding interpol.jl, hyperbolic1D.jl and edit on Operators.jl --- julia/MOLE.jl/examples/hyperbolic1D.jl | 71 ++++++++++++++++++++++++ julia/MOLE.jl/src/Operators/Operators.jl | 3 +- julia/MOLE.jl/src/Operators/interpol.jl | 40 +++++++++++++ 3 files changed, 113 insertions(+), 1 deletion(-) create mode 100644 julia/MOLE.jl/examples/hyperbolic1D.jl create mode 100644 julia/MOLE.jl/src/Operators/interpol.jl diff --git a/julia/MOLE.jl/examples/hyperbolic1D.jl b/julia/MOLE.jl/examples/hyperbolic1D.jl new file mode 100644 index 000000000..0c60354ee --- /dev/null +++ b/julia/MOLE.jl/examples/hyperbolic1D.jl @@ -0,0 +1,71 @@ +#= +Julia implementation of the 1D Advection equation with periodic boundary conditions +from MATLAB MOLE example +=# + +using Printf +using Plots +import MOLE: Operators, BCs + +# Domain limits +west = 0; +east = 1; + +a = 1; #velocity + + +k = 2; #Operator's order of accuracy +m = 50; # number of cells +dx = (east - west) / m; + +t = 1; #Simulation time +dt = dx/abs(a); #CFL condition for explicit schemes + +D = Operators.div(k,m,dx); #1D Mimetic divergence operator +I = Operators.interpol(m, 0.5); #1D 2nd order interpolator + +# 1D Staggered grid +grid = [west, west+dx/2:dx:east-dx/2, east]; + +# Initial Condition +q = 2 * π .* grid +println(typeof(q)) +U = sin.(q)'; + +#Periodic Boundary Condition imposed on the divergence operator +D[1, 2] = 1/(2*dx); +D[1, end-1] = -1/(2*dx); +D[end, 1] = -1/(2*dx); +D[end, end-1] = -1/(2*dx); + +#Premultiply out of the time loop (since it does not change) +D = -a*dt*2*D*I; + +#= + We could have also defined + D = -a*dt*2*I*D + if the grid was defined as: + grid = west : dx :east (nodal) +=# + +U2 = U + D/2*U #Compute one step using Euler's method + + +#Time integration loop +for i in 1: t/dt + #Plot approximation + plot(grid, U2, label="Approximated") + #Plot exact solution + plot!(grid, sin.(2*π .*(grid - a*i*dt)), label="exact") + #Plot attributes + title!("1D Advection Equation with Periodic BC"); + xlabel!("x"); + ylabel!("u(x,t)"); + axis!([west east -1.5 1.5]); + sleep(0.04); + U3 = U + D*U2; #Compute next step using leapfrog scheme + U = U2; + U2 = U3; +end + + diff --git a/julia/MOLE.jl/src/Operators/Operators.jl b/julia/MOLE.jl/src/Operators/Operators.jl index 20491d46b..7af4cfb75 100644 --- a/julia/MOLE.jl/src/Operators/Operators.jl +++ b/julia/MOLE.jl/src/Operators/Operators.jl @@ -3,5 +3,6 @@ module Operators include("gradient.jl") include("divergence.jl") include("laplacian.jl") +include("interpol.jl") -end # module Operators \ No newline at end of file +end # module Operators diff --git a/julia/MOLE.jl/src/Operators/interpol.jl b/julia/MOLE.jl/src/Operators/interpol.jl new file mode 100644 index 000000000..68c9d401c --- /dev/null +++ b/julia/MOLE.jl/src/Operators/interpol.jl @@ -0,0 +1,40 @@ +using SparseArrays +#= + SPDX-License-Identifier: GPL-3.0-or-later + © 2008-2024 San Diego State University Research Foundation (SDSURF). + See LICENSE file or https://www.gnu.org/licenses/gpl-3.0.html for details. +=# + +function interpol(m::Int,c::Float64) + """ + implementation of MATLAB interpol.m operator for Julia + INPUTS: + 'm::Int' : number of cells + 'c::Float' : left interpolation coefficient + OUTPUTS: + I : a (m+1)×(m+2) one-dimensional interpolator of 2nd-order + """ + + #Assertions: + @assert m>=4 ["m >= 4"]; + @assert c >= 0 && c <= 1 ["0 <= c <= 1"]; + + #Dimensions of I: + n_rows = m+1; + n_cols = m+2; + + I = spzeros(n_rows, n_cols); + + I[1,1] = 1; + I[end,end]=1; + + #Average between two continuous cells + avg = [c 1-c]; + + j = 2; + for i = 2: n_rows - 1 + I[i, j:j+2-1] = avg; + j = j + 1; + end +end + From b27d4f0c907d9d2b6cb8ce8e987c05001ad19630 Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Sat, 2 May 2026 23:49:22 -0700 Subject: [PATCH 02/15] Completed hyperbolic1D and work in progress on burgers1D --- julia/MOLE.jl/examples/burgers1D.jl | 53 ++++++++++++++++++++++++++ julia/MOLE.jl/examples/hyperbolic1D.jl | 33 ++++++++-------- 2 files changed, 69 insertions(+), 17 deletions(-) create mode 100644 julia/MOLE.jl/examples/burgers1D.jl diff --git a/julia/MOLE.jl/examples/burgers1D.jl b/julia/MOLE.jl/examples/burgers1D.jl new file mode 100644 index 000000000..83907a07f --- /dev/null +++ b/julia/MOLE.jl/examples/burgers1D.jl @@ -0,0 +1,53 @@ +#= +Solve the 1D Inviscid Burgers' equation +Upwind Scheme is used and the equation is written in conservative form. +Initial condition: exp(-x^2/50) +=# + +using Trapz +using Plots +import MOLE: Operators, BCs + +#Domain limits +west = -15; +east = 15; + +k = 2; #Operator's order of accuracy +m = 300; #Number of cells +dx = (east - west)/m; + +t = 10; #Simulation time +dt = dx; #CFL condition for explicit schemes + +D = Operators.div(k,m,dx) #1D Mimetic divergence operator +I = Operators.interpol(m,1.0) #1D interpolator + +#Use I = Operators.interpol(m,0) (downwind) if the fluid propagates to the left + +#1D Staggered grid +xgrid = [west; (west+(dx/2)):dx: (east-(dx/2)); east]; + +#Initial Condition +U = exp.(-(xgrid.^2)/50); +println(size(U)) +#Premultiply out of the time loop (since it does not change) +D = -dt/2*D*I; +println(size(D)) + +vx = range(west,east, length=k+m) +#Time integration loop +for i in 0: t/dt + #Check for area conservation + trapz(vx,U) + #Plot approximation + plot(xgrid, U, label="Approximated", lw=2) + #Plot attributes + title!("1D Inviscid Burgers' Equation t = $(round(i*dt, sigdigits=2))"); + xlabel!("x"); + ylabel!("u(x,t)"); + plot!(legend=:topleft) + sleep(0.01) + U2 = U + D * U.^2; + global U = U2; + gui() +end diff --git a/julia/MOLE.jl/examples/hyperbolic1D.jl b/julia/MOLE.jl/examples/hyperbolic1D.jl index 0c60354ee..e0641651e 100644 --- a/julia/MOLE.jl/examples/hyperbolic1D.jl +++ b/julia/MOLE.jl/examples/hyperbolic1D.jl @@ -1,9 +1,8 @@ #= Julia implementation of the 1D Advection equation with periodic boundary conditions -from MATLAB MOLE example +from MATLAB MOLE example using the Leapfrog Scheme =# -using Printf using Plots import MOLE: Operators, BCs @@ -25,21 +24,19 @@ D = Operators.div(k,m,dx); #1D Mimetic divergence operator I = Operators.interpol(m, 0.5); #1D 2nd order interpolator # 1D Staggered grid -grid = [west, west+dx/2:dx:east-dx/2, east]; +grid = [west; (west+(dx/2)):dx: (east-(dx/2)); east]; # Initial Condition -q = 2 * π .* grid -println(typeof(q)) -U = sin.(q)'; +U = sin.(2 * π * grid); #Periodic Boundary Condition imposed on the divergence operator D[1, 2] = 1/(2*dx); D[1, end-1] = -1/(2*dx); -D[end, 1] = -1/(2*dx); +D[end, 2] = 1/(2*dx); D[end, end-1] = -1/(2*dx); #Premultiply out of the time loop (since it does not change) -D = -a*dt*2*D*I; +D = -a*dt*2 *D*I; #= We could have also defined @@ -48,24 +45,26 @@ D = -a*dt*2*D*I; grid = west : dx :east (nodal) =# -U2 = U + D/2*U #Compute one step using Euler's method - +U2 = U + D/2*U; #Compute one step using Euler's method #Time integration loop for i in 1: t/dt #Plot approximation - plot(grid, U2, label="Approximated") + plot(grid, U2, label="Approximated", ls=:dot, lw=3) #Plot exact solution - plot!(grid, sin.(2*π .*(grid - a*i*dt)), label="exact") + plot!(grid, sin.(2*π .*(grid .- a*i*dt)), label="exact") #Plot attributes - title!("1D Advection Equation with Periodic BC"); + title!("1D Advection Equation with Periodic BC t = $(round(i*dt, sigdigits=2))"); xlabel!("x"); ylabel!("u(x,t)"); - axis!([west east -1.5 1.5]); + xlims!(west, east); + ylims!(-1.5, 1.5); + plot!(legend=:bottomright) sleep(0.04); - U3 = U + D*U2; #Compute next step using leapfrog scheme - U = U2; - U2 = U3; + U3 = U + D* U2; #Compute next step using leapfrog scheme + global U = U2; + global U2 = U3; + gui() end From f2bcaac184c33f8b76cdd84ab2ff7fa4884bf2ba Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Sat, 2 May 2026 23:53:32 -0700 Subject: [PATCH 03/15] modified changes on interpol.jl --- julia/MOLE.jl/src/Operators/interpol.jl | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/julia/MOLE.jl/src/Operators/interpol.jl b/julia/MOLE.jl/src/Operators/interpol.jl index 68c9d401c..9d47b74cc 100644 --- a/julia/MOLE.jl/src/Operators/interpol.jl +++ b/julia/MOLE.jl/src/Operators/interpol.jl @@ -23,7 +23,7 @@ function interpol(m::Int,c::Float64) n_rows = m+1; n_cols = m+2; - I = spzeros(n_rows, n_cols); + I = zeros(n_rows, n_cols); I[1,1] = 1; I[end,end]=1; @@ -32,9 +32,10 @@ function interpol(m::Int,c::Float64) avg = [c 1-c]; j = 2; - for i = 2: n_rows - 1 + for i in 2: n_rows - 1 I[i, j:j+2-1] = avg; j = j + 1; end + return I end From e26298cbc7f4b38bdee8b29b9b4bb74f80deaa7a Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Sun, 3 May 2026 00:00:30 -0700 Subject: [PATCH 04/15] removed comments print tstatements from burgers1D.jl --- julia/MOLE.jl/examples/burgers1D.jl | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/julia/MOLE.jl/examples/burgers1D.jl b/julia/MOLE.jl/examples/burgers1D.jl index 83907a07f..fab85fa01 100644 --- a/julia/MOLE.jl/examples/burgers1D.jl +++ b/julia/MOLE.jl/examples/burgers1D.jl @@ -29,10 +29,9 @@ xgrid = [west; (west+(dx/2)):dx: (east-(dx/2)); east]; #Initial Condition U = exp.(-(xgrid.^2)/50); -println(size(U)) + #Premultiply out of the time loop (since it does not change) D = -dt/2*D*I; -println(size(D)) vx = range(west,east, length=k+m) #Time integration loop From 33e5605fa365e5ea87181f9eda9c7cd7c199e4c8 Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Sun, 3 May 2026 01:18:46 -0700 Subject: [PATCH 05/15] Added tests for interpol operator --- julia/MOLE.jl/test/Operators/interpol.jl | 43 ++++++++++++++++++++++++ 1 file changed, 43 insertions(+) create mode 100644 julia/MOLE.jl/test/Operators/interpol.jl diff --git a/julia/MOLE.jl/test/Operators/interpol.jl b/julia/MOLE.jl/test/Operators/interpol.jl new file mode 100644 index 000000000..6f2742c21 --- /dev/null +++ b/julia/MOLE.jl/test/Operators/interpol.jl @@ -0,0 +1,43 @@ +tol = 1e-10 +#= +@testset "Testing interpolating operator for m=5 and c=0.5" begin + m = 5 + c = 0.5 + I = Operators.interpol(m,c) + A = [1 0 0 0 0 0 0; + 0 0.5 0.5 0 0 0 0; + 0 0 0.5 0.5 0 0 0; + 0 0 0 0.5 0.5 0 0; + 0 0 0 0 0.5 0.5 0; + 0 0 0 0 0 0 1] +@test norm(I-A) < tol +end + + +@testset "Testing interpolating operator for m=5 and c=1" begin + m = 5 + c = 1.0 + I = Operators.interpol(m,c) + A = [1 0 0 0 0 0 0; + 0 1 0 0 0 0 0; + 0 0 1 0 0 0 0; + 0 0 0 1 0 0 0; + 0 0 0 0 1 0 0; + 0 0 0 0 0 0 1] +@test norm(I-A) < tol +end +=# + +@testset "Testing interpolator operator for m=5 and c=$c" for c=0:0.25:1 + m = 5 + I = Operators.interpol(m,c) + A = [1 0 0 0 0 0 0; + 0 c 1-c 0 0 0 0; + 0 0 c 1-c 0 0 0; + 0 0 0 c 1-c 0 0; + 0 0 0 0 c 1-c 0; + 0 0 0 0 0 0 1] +@test norm(I-A) < tol +end + + From 41f7396e60c6c227137211a879a2ee7476e4b97f Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Sun, 3 May 2026 01:19:47 -0700 Subject: [PATCH 06/15] adding interpol test --- julia/MOLE.jl/test/runtests.jl | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/julia/MOLE.jl/test/runtests.jl b/julia/MOLE.jl/test/runtests.jl index 037ea604e..46d705942 100644 --- a/julia/MOLE.jl/test/runtests.jl +++ b/julia/MOLE.jl/test/runtests.jl @@ -14,6 +14,10 @@ import MOLE: Operators, BCs @testset "Testing 1D laplacian" begin include("Operators/laplacian.jl") end + + @testset "Testing interpol" begin + include("Operators/interpol.jl") + end end @testset "Testing Boundary Conditions" begin From 51d382d8a8a70bc04611c01d0f233ce1c32b92be Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Tue, 19 May 2026 16:03:18 -0700 Subject: [PATCH 07/15] Created function method for burgers, and added png and animation --- .../MOLE.jl/examples/hyperbolic/burgers1D.jl | 106 ++++++++++++++++++ 1 file changed, 106 insertions(+) create mode 100644 julia/MOLE.jl/examples/hyperbolic/burgers1D.jl diff --git a/julia/MOLE.jl/examples/hyperbolic/burgers1D.jl b/julia/MOLE.jl/examples/hyperbolic/burgers1D.jl new file mode 100644 index 000000000..8e1a73551 --- /dev/null +++ b/julia/MOLE.jl/examples/hyperbolic/burgers1D.jl @@ -0,0 +1,106 @@ +#= +Solve the 1D Inviscid Burgers' equation +Upwind Scheme is used and the equation is written in conservative form. +Initial condition: exp(-x^2/50) +=# + +using Plots +import MOLE: Operators, BCs + + +function upwind_burgers(u0, xgrid, T, k, m, dx) +#Solves Burgers equation using MOLE Operators + #= + INPUTS: + u0 : Initial condition + xgrid : staggered grid + T : final time + k : Operator order of accuarcy + m : number of cells + OUTPUTS: + U : solution + t : time interval + =# + #CFL Condition for explicit schemes + dt = dx; + + #Create stepsize for time with given T + t = collect(0.0:dt:T) + + #Use of MOLE Operators + D = Operators.div(k, m, dx) + I = Operators.interpol(m, 1.0) + + #Premultiply out of time loop (does not change) + D = -dt/2*D*I + + #Create an array that holds solution at each time step + U = zeros(length(t)+1,length(xgrid)) + + #Set initial condition at first time element + U[1,:] .= u0.(xgrid) + + for k in eachindex(t) + U[k+1,:] .= U[k,:] + D * U[k,:].^2; + end + + return t, U + +end + +### MAIN LOOP ### + +#Domain limits +west = -15; +east = 15; + +#number of cells +m = 300; + +#stepsize +dx = (east - west)/m; + +#Simulation time +T = 13 + +#Operator's order of accuracy +k = 2; + +# 1D Staggered grid +xgrid = [west; (west + (dx/2)):dx: (east - (dx/2)); east]; + +# Initial Condition +u0(x) = exp.(-(x.^2)/50); + +t, U = upwind_burgers(u0, xgrid, T, k, m, dx) + +path = joinpath(@__DIR__, "output") # Output path to store generated plots +mkpath(path) + + +animation = Plots.@animate for k in eachindex(t) + Plots.plot(xgrid, U[k,:]; + label = "approximated", + xlabel = "x", + ylabel = "u(x,t)", + grid = true, + title = "1D Inviscid Burgers' Equation t = $(round(t[k]; digits=3))", + legend = :topleft + ) +end + + +Plots.gif(animation, joinpath(path,"burgers1D.gif"), fps=10) + +Plots.png( + Plots.plot(xgrid, U[end,:]; + label = "approximated", + xlabel = "x", + ylabel = "u(x,t)", + grid = true, + title = "1D Inviscid Burgers' Equation t = $(round(t[end]; digits=3))", + legend = :topleft + ), + joinpath(path,"burgers_end.png"), +) + From 37d92044fb161544e164faf93ab2031eea689507 Mon Sep 17 00:00:00 2001 From: Jose Luis Baranda Date: Tue, 19 May 2026 16:04:13 -0700 Subject: [PATCH 08/15] png and gif of Burgers1D --- .../examples/hyperbolic/output/burgers1D.gif | Bin 0 -> 943543 bytes .../examples/hyperbolic/output/burgers_end.png | Bin 0 -> 23904 bytes 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 julia/MOLE.jl/examples/hyperbolic/output/burgers1D.gif create mode 100644 julia/MOLE.jl/examples/hyperbolic/output/burgers_end.png diff --git a/julia/MOLE.jl/examples/hyperbolic/output/burgers1D.gif b/julia/MOLE.jl/examples/hyperbolic/output/burgers1D.gif new file mode 100644 index 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