High Order Accurate Hermite Schemes on Curvilinear Grids with Compatibility Boundary Conditions

TL;DR

This work advances high-order Hermite methods for the wave equation on curvilinear grids by integrating compatibility boundary conditions (CBCs) to achieve centered, high-order boundary approximations. It introduces both FOT and SOT Hermite schemes with space-time orders $2m-1$ and $2m$, respectively, and develops CBCs for Cartesian and curvilinear geometries, including corners, through recursive formulations and boundary-derivative constraints. The paper provides solvability and conditioning analyses of the CBC systems, demonstrates symmetry properties on Cartesian grids, and presents comprehensive 2D numerical experiments across multiple grid mappings and boundary types, establishing the schemes’ accuracy and stability. It also discusses practical considerations, conditioning limits (manageable up to $m oughly5$), and outlines directions for future work such as 3D extensions, interfaces, and potential CFL-relaxation via dissipation or filters.

Abstract

High order accurate Hermite methods for the wave equation on curvilinear domains are presented. Boundaries are treated using centered compatibility conditions rather than more standard one-sided approximations. Both first-order-in-time (FOT) and second-order-in-time (SOT) Hermite schemes are developed. Hermite methods use the solution and multiple derivatives as unknowns and achieve space-time orders of accuracy $2m-1$ (FOT) and $2m$ (SOT) for methods using $(m+1)^d$ degree of freedom per node in $d$ dimensions. The compatibility boundary conditions (CBCs) are based on taking time derivatives of the boundary conditions and using the governing equations to replace the time derivatives with spatial derivatives. These resulting constraint equations augment the Hermite scheme on the boundary. The solvability of the equations resulting from the compatibility conditions are analyzed. Numerical examples demonstrate the accuracy and stability of the new schemes in two dimensions.

Figures (27)

• Figure 1: Stages of the Hermite FOT and SOT schemes. Open circles have $(m+1)^2$ degrees of freedom while solid circles have $(2m+2)^2$ degrees of freedom.
• Figure 2: Compatibility boundary conditions, together with interior data, are used to define the Hermite representation on the boundary.
• Figure 3: CBC matrix condition numbers on curvilinear grids using row-scaling (rs) and equilibration (eq) for a Dirichlet BC (D) , Neumann (N) BC, D-D corner, N-N corner, and D-N corner. Top: polynomial mapping. Bottom: X mapping. Left column: $m=1$. Middle column: $m=2$. Right column: $m=3$.
• Figure 4: Plots of the grids used in testing the Hermite schemes. Top, left to right: identity, polynomial and tanh grids. Bottom, left to right: annulus, rhombus and X grids.
• Figure 5: Rhombus. Computed solution and error using the FOT scheme with $m=3$ (order $5$) and the sine solution. Boundary conditions are Dirichlet (left, bottom) and Neumann (right and top).

Theorems & Definitions (3)

• Theorem 1: CBC solvability for Cartesian grids.
• Theorem 2: CBC solvability for Curvilinear grids.
• Theorem 3: Symmetry of the CBC conditions