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Updated: Dec 25, 2025

Creating a Structurally Realistic Finite Element Geometric Model of a Cardiomyocyte to Study the Role of Cellular Architecture in Cardiomyocyte Systems Biology
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Two-Dimensional RBF-ENO Method on Unstructured Grids.

Jan S Hesthaven1, Fabian Mönkeberg1

  • 1SB-MATH-MCSS, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland.

Journal of Scientific Computing
|April 1, 2020
PubMed
Summary
This summary is machine-generated.

We developed a robust high-order Essentially Non-Oscillatory (ENO) method using radial basis functions (RBFs) for solving conservation laws on unstructured grids. This new approach enhances stability and accuracy for complex fluid dynamics simulations.

Keywords:
Euler equationsFinite volume methodsHigh-order methodsRadial basis functionsUnstructured grids

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Area of Science:

  • Computational fluid dynamics
  • Numerical analysis
  • Applied mathematics

Background:

  • Essentially Non-Oscillatory (ENO) and weighted ENO (WENO) methods are standard for solving partial differential equations with discontinuities on structured grids.
  • Stable ENO/WENO methods on unstructured grids remain a significant challenge in computational science.

Purpose of the Study:

  • To propose a novel high-order ENO method utilizing radial basis functions (RBFs) for hyperbolic conservation laws on general 2D unstructured grids.
  • To address the limitations of existing methods in handling ill-conditioned cell geometries and ensuring numerical stability.

Main Methods:

  • Developed an RBF-based reconstruction technique for flexible handling of complex grid structures.
  • Introduced an RBF-based smoothness indicator and a stencil selection algorithm tailored for general meshes.
  • Implemented a stable finite volume evaluation of RBF reconstruction to prevent error stagnation and bound condition numbers.

Main Results:

  • Demonstrated the flexibility of RBF reconstruction for ill-conditioned cell constellations.
  • Successfully developed a stable ENO method applicable to general 2D grids.
  • Validated the method's robustness through challenging numerical examples.

Conclusions:

  • The proposed RBF-based ENO method offers a stable and robust solution for hyperbolic conservation laws on unstructured grids.
  • This work advances the development of high-order numerical methods for complex geometries in computational fluid dynamics.