This paper introduces the basic concepts for physics-compatible discretization techniques. The paper gives a clear distinction between vectors and forms. Based on the difference between forms and pseudo-forms and the -operator which switches between the two, a dual grid description and a single grid description are presented. The dual grid method resembles a staggered finite volume method, whereas the single grid approach shows a strong resemblance with a finite element method. Both approaches are compared for the Poisson equation for volume forms.

In this work we present a mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured grids. The essential ingredients to achieve this are: (i) a velocity-vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular grids.

The research program of the TCV tokamak ranges from conventional to advanced-tokamak scenarios and alternative divertor configurations, to exploratory plasmas driven by theoretical insight, exploiting the device’s unique shaping capabilities. Disruption avoidance by real-time locked mode prevention or unlocking with electron-cyclotron resonance heating (ECRH) was thoroughly documented, using magnetic and radiation triggers. Runaway generation with high-Z noble-gas injection and runaway dissipation by subsequent Ne or Ar injection were studied for model validation. The new 1 MW neutral beam injector has expanded the parameter range, now encompassing ELMy H-modes in an ITER-like shape and nearly non-inductive H-mode discharges sustained by electron cyclotron and neutral beam current drive. In the H-mode, the pedestal pressure increases modestly with nitrogen seeding while fueling moves the density pedestal outwards, but the plasma stored energy is largely uncorrelated to either seeding or fueling. High fueling at high triangularity is key to accessing the attractive small edge-localized mode (type-II) regime. Turbulence is reduced in the core at negative triangularity, consistent with increased confinement and in accord with global gyrokinetic simulations. The geodesic acoustic mode, possibly coupled with avalanche events, has been linked with particle flow to the wall in diverted plasmas. Detachment, scrape-off layer transport, and turbulence were studied in L- and H-modes in both standard and alternative configurations (snowflake, super-X, and beyond). The detachment process is caused by power ‘starvation’ reducing the ionization source, with volume recombination playing only a minor role. Partial detachment in the H-mode is obtained with impurity seeding and has shown little dependence on flux expansion in standard single-null geometry. In the attached L-mode phase, increasing the outer connection length reduces the in–out heat-flow asymmetry. A doublet plasma, featuring an internal X-point, was achieved successfully, and a transport barrier was observed in the mantle just outside the internal separatrix. In the near future variable-configuration baffles and possibly divertor pumping will be introduced to investigate the effect of divertor closure on exhaust and performance, and 3.5 MW ECRH and 1 MW neutral beam injection heating will be added.

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