Small Derivations

This page contains miscellaneous theoretical derivations useful for Gkeyll development, including flux surface averaging in discontinuous Galerkin representation and other computational methods.

Flux Surface Average in DG Representation

Problem Motivation

This derivation addresses the need for a new Dirichlet boundary condition in the Poisson solver at target corners of 3×2v core+SOL simulations. The boundary value must be constant in y-direction to ensure continuity between twist-and-shift boundary conditions (TSBC) and sheath entrance BC

Boundary Condition Options

Zero Dirichlet BC: Equivalent to perfectly conducting wall BC at target corner

• Result: Strong internal transport barrier formation
• Physical interpretation: Unrealistic for most scenarios

Flux Surface Average BC: Set boundary value to flux surface average of electrostatic potential

Flux Surface Average Definition

The flux surface average of the electrostatic potential:

$$⟨φ⟩_{fs}(x) = ∫∫ dz dy J_{xyz} φ(x,y,z) / ∫∫ dz dy J_{xyz}$$

Where $J_{xyz}$ is the Jacobian associated with configuration space coordinates $(x,y,z)$.

In discontinuous Galerkin representation, a configuration space field A is expressed as:

$$A(x,y,z) = Σ_{ijk}^{N_c} Σ_n^{N_b} A_{ijk}^{(n)} φ_{ijk}^{(n)}(x,y,z)$$

Where:

• $N_c$: Number of cells in each direction
• $N_b$: Number of DG basis functions per cell
• $A_{ijk}^{(n)}$: Modal coefficients
• $φ_{ijk}^{(n)}$: DG basis functions

The integral over a surface in DG representation:

$$∫∫ dz dy A(x,y,z) = Σ_{ijk}^{N_c} Σ_n^{N_b} A_{ijk}^{(n)} ∫∫_{c_{jk}} φ_{ijk}^{(n)}(x,y,z) dz dy$$

Where the integration is performed over cell c_{jk}.

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