Inf-Sup Stable Space-Time Discretization of the Wave Equation Based on a First-Order-In-Time Variational Formulation

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Partial Differential Equations

Background:

• The wave equation is fundamental in physics and engineering.
• Existing discretization methods often face stability limitations or mesh restrictions.
• A robust and flexible numerical method is needed for accurate wave propagation simulations.

Purpose of the Study:

• To develop a conforming space-time discretization for the first-order wave equation.
• To extend existing methods by incorporating exponential time weights for enhanced stability.
• To achieve optimal convergence rates without mesh or time-step restrictions.

Main Methods:

• Utilizing a first-order-in-time variational formulation.
• Incorporating exponential weights in the time discretization.
• Applying elliptic projections to derive convergence properties.
• Employing conforming space-time tensor product subspaces.

Main Results:

• Established an inf-sup stability condition for arbitrary discrete subspaces, including splines.
• Achieved optimal convergence rates in energy and L^2 norms for smooth solutions.
• Demonstrated stability independent of mesh size and time step.
• Numerical examples confirmed the theoretical convergence rates.

Conclusions:

• The proposed space-time discretization offers a stable and accurate numerical solution for the wave equation.
• The method's flexibility in choosing discrete subspaces and its independence from mesh/time-step constraints make it broadly applicable.
• This work provides a robust foundation for simulating wave phenomena with high fidelity.