Wave equation

Consider the linear wave equation

\[\begin{aligned} \partial_t^2 u(t,x) &= \partial_x^2 u(t,x), && t \in (0,T), x \in (x_{min}, x_{max}), \\ u(0,x) &= u_0(x), && x \in (x_{min}, x_{max}), \\ \partial_t u(0,x) &= v_0(x), && x \in (x_{min}, x_{max}), \\ \text{boundary conditions}, &&& x \in \partial (x_{min}, x_{max}). \end{aligned}\]

SummationByPartsOperators.jl includes a pre-built semidiscretization of this equation: WaveEquationNonperiodicSemidiscretization. Have a look at the source code if you want to dig deeper. In particular, you can find applications of derivative_left, derivative_right mul_transpose_derivative_left!, and mul_transpose_derivative_right!. Below is an example demonstrating how to use this semidiscretization.

using SummationByPartsOperators, OrdinaryDiffEqTsit5using LaTeXStrings; using Plots: Plots, plot, plot!, savefig# general parametersxmin = -1.xmax = +1.tspan = (0., 8.0)u0_func(x) = exp(-20x^2)v0_func(x) = zero(x)# HomogeneousNeumann, HomogeneousDirichlet, and NonReflecting BCs are availableleft_bc  = Val(:HomogeneousNeumann)right_bc = Val(:HomogeneousDirichlet)# setup spatial semidiscretizationD2 = derivative_operator(MattssonSvärdShoeybi2008(), derivative_order = 2,                         accuracy_order = 4, xmin = xmin, xmax = xmax, N = 101)semi = WaveEquationNonperiodicSemidiscretization(D2, left_bc, right_bc)ode = semidiscretize(v0_func, u0_func, semi, tspan)# solve second-order ODE using a Runge-Kutta methodsol = solve(ode, Tsit5(), saveat = range(first(tspan), stop = last(tspan), length = 200))# visualize the resultplot(xguide = L"x")plot!(evaluate_coefficients(sol.u[end].x[2], semi), label = L"u")plot!(evaluate_coefficients(sol.u[end].x[1], semi), label = L"\partial_t u")savefig("example_wave_equation.png");

"/home/runner/work/SummationByPartsOperators.jl/SummationByPartsOperators.jl/docs/build/tutorials/example_wave_equation.png"

Advanced visualization of different boundary conditions

Let's create animations of the numerical solutions for different boundary conditions.

using Printf; using Plots: Animation, frame, giffunction create_gif(left_bc::Val{LEFT_BC}, right_bc::Val{RIGHT_BC}) where {LEFT_BC, RIGHT_BC}    xmin = -1.    xmax = +1.    tspan = (0., 8.0)    u0_func(x) = exp(-20x^2)    v0_func(x) = zero(x)    D2 = derivative_operator(MattssonSvärdShoeybi2008(), derivative_order = 2,                            accuracy_order = 4, xmin = xmin, xmax = xmax, N = 101)    semi = WaveEquationNonperiodicSemidiscretization(D2, left_bc, right_bc)    ode = semidiscretize(v0_func, u0_func, semi, tspan)    sol = solve(ode, Tsit5(), saveat=range(first(tspan), stop=last(tspan), length=200))    anim = Animation()    idx = 1    x, u = evaluate_coefficients(sol.u[idx].x[2], D2)    fig = plot(x, u, xguide = L"x", yguide = L"u", xlim = extrema(x), ylim = (-1.05, 1.05),              label = "", title = @sprintf("\$t = %6.2f \$", sol.t[idx]))    for idx in 1:length(sol.t)        fig[1] = x, sol.u[idx].x[2]        plot!(title = @sprintf("\$t = %6.2f \$", sol.t[idx]))        frame(anim)    end    gif(anim, "wave_equation_$(LEFT_BC)_$(RIGHT_BC).gif")endcreate_gif(Val(:HomogeneousNeumann), Val(:HomogeneousNeumann))

create_gif(Val(:HomogeneousNeumann), Val(:HomogeneousDirichlet))

create_gif(Val(:HomogeneousNeumann), Val(:NonReflecting))

Package versions

These results were obtained using the following versions.

using InteractiveUtilsversioninfo()using PkgPkg.status(["SummationByPartsOperators", "OrdinaryDiffEqTsit5"],           mode = PKGMODE_MANIFEST)

Julia Version 1.10.12
Commit d93beab124c (2026-08-15 10:29 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 9V45 96-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, generic)
Threads: 2 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
Status `~/work/SummationByPartsOperators.jl/SummationByPartsOperators.jl/docs/Manifest.toml`
  [b1df2697] OrdinaryDiffEqTsit5 v2.1.4
  [9f78cca6] SummationByPartsOperators v0.5.97 `~/work/SummationByPartsOperators.jl/SummationByPartsOperators.jl`