Structure aware Runge-Kutta time stepping for spacetime tents.

Area of Science:

• Computational physics
• Numerical analysis
• Astrophysics

Background:

• Standard Runge-Kutta methods face challenges with hyperbolic solutions in complex spacetimes.
• High-order spatial discretizations, like discontinuous Galerkin, require compatible time-stepping schemes for optimal accuracy.
• Propagating hyperbolic solutions in tent-shaped spacetime regions necessitates specialized numerical techniques.

Purpose of the Study:

• Introduce a novel class of Runge-Kutta type methods for time stepping hyperbolic solutions.
• Ensure these new methods achieve expected convergence properties with high-order spatial discretizations.
• Demonstrate the effectiveness of the proposed methods through numerical examples.

Main Methods:

• Derivation of nonstandard order conditions tailored for the new Runge-Kutta methods.
• Application of the methods to nonlinear hyperbolic systems for convergence rate analysis.
• Investigation of discrete stability properties for linear hyperbolic equations.

Main Results:

• The new Runge-Kutta methods exhibit optimal convergence rates when combined with high-order spatial discretizations.
• Numerical examples confirm the theoretical predictions for nonlinear hyperbolic systems.
• The study reports on the discrete stability characteristics of the proposed methods.

Conclusions:

• The developed Runge-Kutta methods are suitable for accurately time-stepping hyperbolic solutions in tent-shaped spacetimes.
• These methods overcome limitations of standard approaches, offering improved performance with advanced spatial discretizations.
• The findings contribute to more robust and accurate numerical simulations in computational physics and related fields.