Problems

A central object in WaterWaves1D.jl is the Problem structure, which contains all information on a numerically discretized initial-value problem. In practice, a problem is generated as

problem = Problem( model, initial, times ; solver, label )

where

• model is related to the (spatially discretized) equation at stake. Built-in models are typically generated as

model = MyModel( param ; kwargs )

where param is a NamedTuple containing relevant parameters of the model and of the spatial grid, and kwargs are some optional arguments allowing some choices in the discretization (for instance dealiasing), and a label for future references.

• inital is the couple of initial data. It can be generated for instance using the function Init as

initial = Init( η, v )

where η and v are two functions (representing respectively the surface deformation and the derivative of the trace of the velocity potential at the surface). Alternatively, it can also be built from the values of these functions at equally-spaced collocation points.

• times contains relevant parameters of the time integration: in particular the final time T and the time-step dt. It can be a NamedTuple with these informations or generated via the function Times.
• optionally, the time-solver solver can be provided (built-in solvers are the explicit Euler solver, Euler and Euler_naive, a symplectic Euler solver, EulerSymp, and the explicit Runge-Kutta 4 solver, RK4 and RK4_naive). By default the RK4 solver is used.
• optionally, a string label can be provided for future reference. It is inferred from the model if not provided.

Once it has been built, problem contains the raw data, data (initially just the initial data), in addition to model, initial, times, solver and label. The initial-value problem is then numerically integrated (filling data) with

solve!(problem)