Initial Value Problems

Use InitialValueProblem to evolve fields from an initial state. Construct an InitialValueSolver with a timestepper, then advance it with step! or run!.

Formulation

Write the semidiscrete system as $M\partial_t X+LX=F(X,t)$. For IMEX stepping, put the time derivative and the linear terms to be treated implicitly on the left; put the explicit terms on the right. Register coefficients and expression substitutions with add_parameters! before naming them in equations.

A bounded direction also needs boundary conditions and appropriate tau unknowns. Those constraints make the system a DAE; an initial condition should satisfy them. Fully periodic diffusion, as below, needs no wall BCs or boundary taus.

Example: periodic diffusion

This serial CPU example evolves $u(x,0)=\sin x$ under $\partial_t u=\nu\partial_x^2u$ and checks $u(x,t)=e^{-\nu t}\sin x$.

using Tarang

coords = CartesianCoordinates("x")
dist = Distributor(coords; dtype=Float64, device=CPU())
xb = RealFourier(coords["x"]; size=32, bounds=(0.0, 2π))
domain = Domain(dist, (xb,))
u = ScalarField(domain, "u")
x, = local_grids(dist, xb)
ensure_layout!(u, :g)
get_grid_data(u) .= sin.(x)

problem = InitialValueProblem([u])
add_parameters!(problem; nu=0.1)
add_equation!(problem, "dt(u) - nu*Δ(u) = 0")
solver = InitialValueSolver(problem, RK222(); dt=0.01)
run!(solver; stop_iteration=10, progress=false)

ensure_layout!(u, :g)
expected = exp(-0.1 * solver.sim_time) .* sin.(x)
@assert maximum(abs.(get_grid_data(u) .- expected)) < 1e-8

Stepping and output

step!(solver) uses solver.dt; step!(solver, dt) also updates the stored step size. run! manages stopping conditions, CFL control, callbacks, and registered output handlers. The updated solution remains in the original fields; solver.sim_time and solver.iteration record progress.

Choose the timestepper and backend together using the execution-support table. See Analysis & Output for diagnostics and Running with MPI for distributed setup.

Bounded domains and tutorials

• Tau method: augmented equations and stage constraints.
• Boundary conditions: fixed and moving walls.
• 2D Rayleigh–Bénard convection: coupled velocity, pressure, and temperature.
• 3D turbulence: a periodic flow simulation.
• Surface dynamics and rotating shallow water.

See equation syntax and solver reference for shared APIs.