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Curvilinear High-Order Mimetic Differences that Satisfy Conservation Laws

Angel Boada, Johnny Corbino, Miguel Dumett, Jose Castillo

TL;DR

This paper extends high-order mimetic difference operators (Corbino-Castillo) to curvilinear staggered grids, proving a discrete extended Gauss-Divergence theorem in curved coordinates and establishing energy and mass conservation for the 2D/3D acoustic wave equations. The approach leverages Kronecker-product constructions and curvilinear mappings to build $D$ and $G$ operators with diagonal inner products, maintaining conservative properties through a mapped Jacobian framework with $J_D$, $J_G$, and boundary operator $B$. The authors verify the method via steady-state Poisson tests in semi-annular and sinusoidal regions and a 2D acoustic wave example in a semi-annulus, achieving 4th-order accuracy in curved geometries. These results enable accurate, structure-preserving simulations on nontrivial geometries and lay groundwork for extensions to overlapping grids and embedded boundaries.

Abstract

We investigate the construction and usage of mimetic operators in curvilinear staggered grids. Specifically, we extend the Corbino-Castillo operators so they can be utilized to solve problems in non-trivial geometries. We prove that the resulting curvilinear operators satisfy the discrete analog of the extended Gauss-Divergence theorem. In addition, we demonstrate energy and mass conservation in curvilinear coordinates for the acoustic wave equation. These findings are illustrated in two-dimensional and three-dimensional elliptic/hyperbolic equations and can be extended to other partial differential equations as well.

Curvilinear High-Order Mimetic Differences that Satisfy Conservation Laws

TL;DR

This paper extends high-order mimetic difference operators (Corbino-Castillo) to curvilinear staggered grids, proving a discrete extended Gauss-Divergence theorem in curved coordinates and establishing energy and mass conservation for the 2D/3D acoustic wave equations. The approach leverages Kronecker-product constructions and curvilinear mappings to build and operators with diagonal inner products, maintaining conservative properties through a mapped Jacobian framework with , , and boundary operator . The authors verify the method via steady-state Poisson tests in semi-annular and sinusoidal regions and a 2D acoustic wave example in a semi-annulus, achieving 4th-order accuracy in curved geometries. These results enable accurate, structure-preserving simulations on nontrivial geometries and lay groundwork for extensions to overlapping grids and embedded boundaries.

Abstract

We investigate the construction and usage of mimetic operators in curvilinear staggered grids. Specifically, we extend the Corbino-Castillo operators so they can be utilized to solve problems in non-trivial geometries. We prove that the resulting curvilinear operators satisfy the discrete analog of the extended Gauss-Divergence theorem. In addition, we demonstrate energy and mass conservation in curvilinear coordinates for the acoustic wave equation. These findings are illustrated in two-dimensional and three-dimensional elliptic/hyperbolic equations and can be extended to other partial differential equations as well.
Paper Structure (13 sections, 37 equations, 10 figures, 4 tables)

This paper contains 13 sections, 37 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mimetic Operators
  3. Discrete Extended Gauss-Divergence Theorem in Curvilinear Coordinates
  4. Acoustic Wave Problem and Curvilinear Coordinates
  5. Acoustic Wave Problem and Curvilinear Coordinates
  6. Semi-discrete Energy Conservation
  7. Semi-discrete Mass Conservation
  8. Steady State Numerical Examples
  9. 2D Poisson Equation on a Semi-Annular Region
  10. 2D Poisson equation on a sinusoidal region
  11. 3D Poisson Equation in a Sinusoidal Volume
  12. Acoustic Wave Numerical Example
  13. Concluding Remarks

Figures (10)

  • Figure 1: A 2D uniform staggered grid with $5$ horizontal cells and $3$ vertical cells. Scalar values (), and vector components $\langle u, v \rangle$ are defined at horizontal middle cell faces and vertical middle cell faces, respectively. Observe that discrete scalar and vector fields are not defined at nodes.
  • Figure 2: Exact solution for 2D Poisson equation on semi-annular region.
  • Figure 3: Computed solution for 2D Poisson equation on semi-annular region.
  • Figure 4: Grid in domain with curvilinear boundaries.
  • Figure 5: Exact solution for 2D Poisson equation with curvilinear boundaries.
  • ...and 5 more figures