Physics-driven proper orthogonal decomposition: A simulation methodology for partial differential equations.

Area of Science:

• Computational physics
• Scientific computing
• Numerical analysis

Background:

• Solving partial differential equations (PDEs) often requires significant computational resources.
• Direct Numerical Simulations (DNS) provide accurate solutions but are computationally expensive.
• Model reduction techniques are sought to accelerate simulations.

Purpose of the Study:

• To present a novel simulation methodology based on Proper Orthogonal Decomposition (POD) and Galerkin projection.
• To develop a physics-driven approach for solving PDEs efficiently.
• To demonstrate the methodology's effectiveness on complex physical problems.

Main Methods:

• A learning algorithm based on Proper Orthogonal Decomposition (POD) is employed.
• Solution data from DNSs are used to train POD modes.
• Galerkin projection is applied to the PDE within the POD space.
• The methodology involves data collection, POD mode calculation, and model derivation.

Main Results:

• The POD-Galerkin methodology enables a significant reduction in degrees of freedom (DoF).
• High accuracy is maintained despite the reduction in DoF.
• Computational effort is drastically decreased compared to DNS.
• Successful simulations were performed for dynamic thermal analysis of microprocessors and the Schrödinger equation.

Conclusions:

• The physics-driven POD-Galerkin method offers a computationally efficient alternative to DNS.
• This approach effectively solves complex PDEs by reducing model complexity.
• The methodology shows promise for accelerating simulations in various physical domains.