2D Navier-Stokes Equations Solver for 2D Open Channel Flow

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Introduction

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• This solver, developed in Fortran 2008, solves the 2D Navier-Stokes Equations for a fluid flowing in an Open-Channel which can be thought of as a river.
• The Finite Volume Method (FVM) has been used to discretisise the 2D Navier-Stokes Equations.
• The solver has been parallelised using coarrays and the resulting Algebraic Equations from discretisation are solved via the Successive Over Relaxation Method (SOR).
• The problem-domain is divided up using Domain Decomposition.

Theory

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Domain Decomposition

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• This is a computational technique used to solve large-scale problems - especially partial differential equations (PDEs) — by breaking down the computational domain into smaller subdomains.
• Each subdomain is then solved (often in parallel), and the results are combined to obtain the global solution.

Key Concepts

1.Partitioning:

• The overall problem domain (e.g., a mesh in FEM or grid in FDM) is divided into smaller parts. This can be done geometrically or based on graph partitioning.

2. Subdomain Solvers:

• Each subdomain is treated as an independent problem, often solved using local solvers.
• Communication between neighboring subdomains ensures consistency across interfaces.

3. Parallelisation:

• Domain decomposition is widely used in parallel computing, as each subdomain can be assigned to a different processor or thread.

4. Coupling:

• Solutions at the boundaries of subdomains must be matched (e.g., using iterative methods like Schwarz methods or Lagrange multipliers) to ensure global consistency.

5. Types:

• Overlapping (e.g., Schwarz methods): Subdomains may overlap; iterative data exchange improves convergence.
• Non-overlapping (e.g., Schur complement methods, FETI, BDDC): Subdomains meet at interfaces, requiring global interface solvers.

Requirements

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• Compiler: gfortran.
• OS: Developed and test on Ubuntu 20.04.
• Make to build the software.

Getting and Running the Software

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References

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• Modern Fortran: Building Efficient Parallel Applications by Milan Curcic.

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