Advances in Structure-Preserving and High-Order Numerical Methods

Advances in Numerical Methods and Computational Models

Recent developments in the field of numerical methods and computational models have seen significant advancements, particularly in the areas of structure-preserving algorithms, high-order accuracy schemes, and innovative applications in fluid dynamics and biomedical simulations. The focus has been on creating robust, efficient, and accurate methods that can handle complex systems with dissipative properties, non-linear dynamics, and high-dimensionality.

Structure-Preserving Algorithms: There has been a notable shift towards developing numerical methods that preserve the intrinsic structures of the systems they model, such as symplectic integrators for dissipative systems. These methods not only enhance stability but also improve long-term accuracy, making them suitable for complex dynamical systems like the Navier-Stokes equations.

High-Order Accuracy Schemes: The pursuit of higher-order accuracy in both time and space discretizations has led to the creation of novel schemes that combine predictor-corrector approaches with energy correction methods. These schemes ensure not only mass conservation and positivity preservation but also energy dissipation, which is crucial for accurately modeling systems like the Keller-Segel equations.

Innovative Applications: Computational models are increasingly being tailored for specific applications, such as simulating thermal laser-tissue interactions in robotic surgery and accelerating cardiovascular simulations through harmonic balance methods. These models leverage finite element methods and spectral discretization to achieve high realism and computational efficiency.

Noteworthy Developments:

  • The application of symplectic integrators to dissipative systems, particularly the Navier-Stokes equations, represents a groundbreaking advancement.
  • The development of energy dissipative, mass-conserving schemes for Keller-Segel equations introduces a new approach to handling complex biological dynamics.
  • The harmonic balance method for cardiovascular simulations significantly reduces computational time while maintaining high accuracy.

Sources

Finite element approximation to the non-stationary quasi-geostrophic equation

Numerical integrations of stochastic contact Hamiltonian systems via stochastic contact Hamilton-Jacobi equation

A new class of energy dissipative, mass conserving and positivity/bound-preserving schemes for Keller-Segel equations

Long-term behaviour of symmetric partitioned linear multistep methods I. Global error and conservation of invariants

Unconditionally stable symplectic integrators for the Navier-Stokes equations and other dissipative systems

Maximum-norm a posteriori error bounds for parabolic equations discretised by the extrapolated Euler method in time and FEM in space

Towards a Physics Engine to Simulate Robotic Laser Surgery: Finite Element Modeling of Thermal Laser-Tissue Interactions

Introducing a Harmonic Balance Navier-Stokes Finite Element Solver to Accelerate Cardiovascular Simulations

DiscoTEX 1.0: Discontinuous collocation and implicit-turned-explicit (IMTEX) integration symplectic, symmetric numerical algorithms with high order jumps for differential equations II: extension to higher-orders of numerical convergence

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