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SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES.
1Lilly Corporate Center, DC 4108, Eli Lilly and Company, Indiana, IN 46285, USA.
Discrete and Continuous Dynamical Systems. Series B
|June 16, 2012
Summary
Two new finite difference methods enhance solving Helmholtz equations on irregular domains and interfaces. These methods improve accuracy and efficiency for complex problems, including incompressible Navier-Stokes equations.
Area of Science:
- Numerical Analysis
- Computational Science
- Partial Differential Equations
Background:
- Solving Helmholtz equations is crucial for applications like incompressible Navier-Stokes solvers.
- Existing methods like the augmented immersed interface method (AIIM) face accuracy issues with large λ values on irregular domains.
- Efficiently handling Helmholtz equations with interfaces remains a challenge.
Purpose of the Study:
- To develop novel finite difference methods for solving Helmholtz equations.
- To address accuracy limitations of existing methods for irregular domains and interfaces.
- To ensure numerical stability and computational efficiency.
Main Methods:
- A level set function approach is used to extend the source term and PDE to a larger domain for irregular domains, improving AIIM accuracy.
- A maximum principle preserving finite difference method is developed for interfaces, utilizing a modified five-point stencil.
- The method is extended for temporal discretized equations with λ inversely proportional to mesh size.
Main Results:
- The proposed method enhances accuracy for Helmholtz equations on irregular domains, especially for large λ.
- The new finite difference method for interfaces preserves the maximum principle, leading to a stable linear system.
- The coefficient matrix satisfies the discrete maximum principle's sign property, enabling efficient multigrid solving.
Conclusions:
- The developed finite difference methods offer significant improvements for solving Helmholtz equations in complex scenarios.
- These methods provide accurate and efficient solutions for irregular domains and interfaces, crucial for fluid dynamics and other applications.
- The techniques pave the way for more robust numerical solvers in scientific computing.
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