Mapped tent pitching schemes for hyperbolic systems
For computational scientists solving hyperbolic PDEs, this provides a practical way to leverage existing explicit schemes on non-uniform spacetime meshes, though the approach is incremental.
This paper develops a mapping technique that enables standard explicit time-stepping methods to be used within tent-shaped spacetime meshes for hyperbolic systems, demonstrated on the acoustic wave equation and Euler system.
A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solving hyperbolic systems using such tent meshes to advance in time have been proposed previously. Such schemes have the ability to advance in time by different amounts at different spatial locations. This paper explores a technique by which standard discretizations, including explicit time stepping, can be used within tent-shaped spacetime domains. The technique transforms the equations within a spacetime tent to a domain where space and time are separable. After detailing techniques based on this mapping, several examples including the acoustic wave equation and the Euler system are considered.