SSFG-TSBC Implementation

In the August 2024 version of gkylzero, twist-and-shift (TS) boundary conditions are applied only to the distribution function. The Poisson solver computes charge density using the zeroth moment of this distribution function, but the smoothing operation does not incorporate ghost cells, effectively omitting TSBC-related information.

Consequences

This leads to several issues:

• Electrostatic potential discontinuity at z = ±π boundaries
• Energy conservation violations
• Nonphysical electron heating in core region
• Parallel velocity instabilities in scrape-off layer

Radial profiles showing numerical instability consequences: electron temperature (top left), ion temperature (top right), electrostatic potential (bottom left), and electron parallel fluid velocity (bottom right) during instability.

Electrostatic potential averaged along y-direction showing progressive breakdown of periodicity during spurious electron heating.

Twist-Shift Operators

The TSBC application involves operators:

• T_LU: Lower to upper boundary transformation
• T_UL: Upper to lower boundary transformation

In infinite resolution limit: T_UL T_LU = T_LU T_UL = I

At finite resolution, interpolation introduces diffusion, breaking this identity.

Boundary Condition Requirements

The updated field φ̃ must satisfy:

T_{UL}(φ̃_L) = φ̃_U\\
T_{LU}(φ̃_U) = φ̃_L

Implementation Methods

Phase 1: One-Side Update

Method: Apply TSBC on lower side only

φ̃_L = T_{LU}(φ_U)\\
φ̃_U = φ_U

Advantages:

• Simple implementation
• Effective at resolving baseline issues
• Maintains TS condition

Disadvantages:

• Asymmetric interpolation error
• Creates discontinuity on one side only
• Potential weak point at LCFS-boundary intersection

Phase 1a TSBC method at upper (left) and lower (right) boundaries in TCV simulation.

Phase 1a Validation

Complete TCV simulation with resolution 48×32×16×16×6 shows:

• Electron heating resolved
• Parallel velocity instability eliminated
• Temperature profiles maintained

Comparison showing Phase 1a TSBC effectiveness in resolving baseline issues.

Skin Surface From Ghost (SSFG)

Concept: Modify modal values in skin cells such that boundary nodal values correspond to updated values while keeping inner nodal values constant.

Features:

• Maintains z-continuity
• Breaks x-continuity at LCFS-boundary intersection
• DG-compatible implementation

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