Report on Current Developments in Numerical Methods for Partial Differential Equations

General Direction of the Field

The field of numerical methods for partial differential equations (PDEs) is witnessing a significant shift towards more efficient, stable, and structure-preserving schemes. Recent developments emphasize the integration of advanced mathematical techniques with computational efficiency to handle complex physical phenomena. Key areas of focus include:

1. Decoupled and Energy-Stable Schemes: There is a growing interest in developing decoupled schemes for coupled systems of PDEs, such as the Cahn-Hilliard-Navier-Stokes equations. These schemes aim to simplify the computational process by decoupling the equations, ensuring stability, and providing optimal error estimates.

2. High-Order and Entropy-Stable Discretizations: The pursuit of high-order accuracy and entropy stability is driving innovations in discretization methods. Techniques like summation-by-parts (SBP) operators and weak Galerkin finite element methods are being refined to achieve higher efficiency and stability on unstructured meshes.

3. Structure-Preserving Methods: Ensuring that numerical methods preserve key physical properties, such as energy conservation, vorticity preservation, and geometric invariants, is becoming a central theme. This includes the development of auto-stabilized methods and Lagrange multiplier approaches to maintain these properties in the numerical solutions.

4. Multiscale and Interface Problems: Advances in handling multiscale and interface problems are crucial for applications in porous media and phase transition dynamics. Techniques that can accurately capture high-contrast and high-frequency phenomena are being developed and analyzed.

5. Shape Optimization and Level-Set Methods: The integration of shape optimization with level-set methods is gaining traction, particularly in the context of discontinuous Galerkin methods. These approaches leverage geometric mesh flexibility and high-order temporal discretizations to efficiently solve complex shape optimization problems.

Noteworthy Developments

• Decoupled Schemes for Cahn-Hilliard-Navier-Stokes Equations: The development of a decoupled, first-order, fully discrete, and unconditionally energy stable scheme for the Cahn-Hilliard-Navier-Stokes equations represents a significant advancement in handling coupled systems.
• Tensor-Product Split-Simplex Summation-By-Parts Operators: The introduction of tensor-product split-simplex SBP operators on triangles and tetrahedra enhances the efficiency of SBP discretizations, offering a more than an order of magnitude improvement in computational time.
• Auto-Stabilized Weak Galerkin Finite Element Methods: The proposed auto-stabilized WG method extends to non-convex elements and maintains optimal order error estimates, marking a significant advancement in stabilizer-free WG methods.

These developments highlight the ongoing efforts to push the boundaries of numerical methods for PDEs, ensuring that they are not only accurate and efficient but also preserve the underlying physical and geometric properties of the problems they aim to solve.