Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order.

Area of Science:

• Computational Fluid Dynamics
• Numerical Analysis
• Scientific Computing

Background:

• High-order numerical methods are crucial for accurately solving complex Partial Differential Equations (PDEs).
• Existing iterative schemes often require careful tuning to balance accuracy and computational cost.
• p-adaptivity, adjusting polynomial order locally, offers potential for efficiency but can be challenging to implement.

Purpose of the Study:

• To develop a new paradigm for efficient p-adaptive arbitrary high-order methods.
• To modify existing iterative schemes to naturally incorporate p-adaptivity.
• To enhance computational efficiency and robustness in solving hyperbolic PDEs.

Main Methods:

• Modification of arbitrary high-order iterative schemes to match iteration accuracy with discretization accuracy.
• Recasting the Arbitrary Derivative (ADER) method as a Deferred Correction (DeC) scheme.
• Integration of a local a posteriori limiter for p-adaptivity and structure preservation.

Main Results:

• The modified methods achieve computational advantages by aligning accuracy across iterations.
• Natural p-adaptivity is achieved by stopping iterations based on defined conditions.
• The approach is easily integrated into existing implementations without compromising parallelization.

Conclusions:

• The proposed framework offers a more efficient formulation for arbitrary high-order methods.
• The method demonstrates robustness and computational efficiency on compressible gas dynamics benchmarks.
• This work facilitates the design of advanced, adaptive numerical solvers for PDEs.