Solving the Equations: Unveiling Challenges and Solutions for PDEs in Simulations

Challenges and Solutions for PDEs in Simulations

Computational fluid dynamics (CFD) involves the solution of Second order partial differential equations (PDEs), principally the conservation laws of mass and linear momentum that govern fluid motion and forces. Closed-form solutions of these PDEs exist only for very simple geometries and flow regimes, e.g. the Hagen-Poiseuille equation for laminar flow of an incompressible, Newtonian fluid in a long pipe of constant cross section.

Mathematical characteristics of higher order PDEs:

• Highest Order: The highest order derivative in a PDE may be continuous or discontinuous over a specific portion of the domain.
• Characteristics: Certain lines or curves, called characteristics, can exist across which discontinuities in the highest order derivative may occur.
• Numerical Strategies: The choice of a suitable numerical approach to solve a PDE depends on its specific nature due to the potential for discontinuities.
• Steady-State Conduction: This elliptic equation represents a boundary value problem. While the boundaries may have discontinuities, the solution itself is always smooth because disturbances propagate instantly in all directions.
• Unsteady Conduction: This problem is parabolic in time and elliptic in space, making it an initial-boundary value problem. The solution can exhibit discontinuities due to the parabolic nature.

In terms of real-world problems, for instance, turbulent flow around a complex geometry such as a car, engineers use CFD to calculate flow velocities and pressures to predict aerodynamic performance (lift and drag forces). Engineers visualize the flow with streamlines, cutting planes, boundary surfaces, and many more.

Fig: Aerodynamics analysis (turbulence) of F1 2021 concept race car with 18 million mesh cells (source:https://fetchcfd.com/view-project/1603-f1-2021-aerodynamics-%7C-cfd-simulation)

However, despite their importance, solving these equations on classical computers can be surprisingly difficult, especially for complex situations such as 2D Burgers.

2D Burgers’ Equation (CFD):

A second order PDE (partial differential equation), such as two-dimensional Burgers’ equation is a fundamental tool in the field of Computational Fluid Dynamics (CFD), used to model the behavior of viscous fluids, an important aspect while modeling simulations.

Despite its seemingly straightforward appearance, the equation’s non-linearity presents significant challenges in obtaining solutions for practical applications.

The challenge in solving PDEs for simulations

Engineers solving 2D Burgers Equations using classical computing hardware and algorithms can face several challenges.

Challenges in solving PDEs with Classical Computers and Algorithms:

1. Discretization:

Classical computers require discretizing the simulation domain (the space being modeled) into a grid of points. The finer the grid, the more accurate the solution, but it comes at a cost:

• Computational Strain: More grid points equate to more calculations, exponentially increasing processing time for complex simulations. More computing cores are required for faster calculations, hence increasing costs.
• Stiffness: The Burgers’ equation can become stiff, meaning numerical errors can rapidly amplify with larger time steps. Designing stable numerical schemes for both accuracy and efficiency requires careful consideration.

2. Boundary conditions:

Defining realistic boundary conditions (how the fluid behaves at edges) becomes complex for real-world scenarios. Implementing these conditions accurately in a discretized grid system can introduce errors or require additional computational resources.

3. Turbulence:

Modeling turbulence, a crucial aspect of many CFD problems, significantly increases the complexity of the Burgers’ equation. It demands advanced numerical approaches that can be computationally expensive.

High-Performance Computing (HPC) and Beyond

While classical computers have limitations, advancements are being made:

• HPC Power: Utilizing parallel computing with multiple processors tackles larger simulations using Quantum Algorithms.
• Adaptive Mesh Refinement: Refining the grid only in regions where higher accuracy is needed, optimizing resource usage.
• Machine Learning Assistance: Emerging techniques use quantum machine learning to accelerate simulations or improve accuracy.

Running Quantum Algorithms on HPCs for solving PDEs used in CFD:

BosonQ Psi is using Quantum algorithms for solving these PDEs, such as the 2D Burgers Equation. It emerges as a potential game-changer for solving specific aspects of PDEs that are successfully delivering higher accuracy, scalability, and speed-up over existing classical-only methods.

Here’s why:

Quantum algorithms (QA):

QA tackles the linear systems that often arise when solving PDEs numerically. It leverages the power of quantum information processing to potentially solve these systems more efficiently for large problems using existing HPCs and Quantum Computers.

Simplified breakdown:

1. Encoding the System: The linear system is translated into a quantum circuit using qubits and quantum operations.
2. Variational approach: The algorithm prepares a quantum state that’s “close” to the solution. It iteratively refines this state by applying controlled operations, guided by a cost function.
3. Measurement and Solution: Finally, the quantum state is measured, providing an approximation to the solution of the original linear system.

Breakthrough innovation with Quantum algorithms in CFD:

• Scalable and more accurate: BQP’s Quantum algorithms scaled more efficiently than classical methods for large systems, with better accuracy than the traditional classical approach. This stems from the power of superposition and quantum parallelization used in quantum algorithms, which allow them to explore multiple solutions simultaneously.
• Adaptability: BQP’s CFD solver, powered by quantum algorithms, can be adapted to different types of linear systems, making it versatile for various applications within Aerospace, automotive, and many more.
• Solving previously unsolvable problems in CFD: Classical computers struggled to model and compute solutions for niche problems in CFD, hindering innovation in crucial areas. Our solutions bridge this gap, offering robust, consistent, and more accurate solutions that can be scaled to tackle larger industrial problems, particularly in aerospace and other areas.

Solving tough partial differential equations (PDEs) for computational fluid dynamics (CFD) with quantum computing algorithms marks a substantial departure from conventional simulation approaches,offering unparalleled scalability, accuracy, and speed.

As systems become more complex, the challenge of solving CFDs with numerous constraints and boundary conditions grows exponentially, impeding innovation and design exploration.

Many problems are so intricate that they cannot even be modeled mathematically for classical simulations. Quantum computing’s inherent parallelism and ability to handle large, complex datasets offer significant speedups over traditional methods, overcoming these constraints.

BQP’s simulation platform, BQPhy, leverages quantum-inspired algorithms to tackle these challenging PDEs for intricate CFD scenarios.

Engineers, simulation enthusiasts and researchers are invited to partner with us to explore the potential of quantum computing in advancing CFD simulations using next generation simulation and design exploration techniques using BQP’s innovative quantum inspired techniques