The Mathematics of the Simulation Workloads
A map of the equation families L0 solves, from transport and MHD to transport PDEs, and the numerical methods each demands.
One foundation, several equation families
The L0 workloads are diverse, but they reduce to a handful of mathematical families, each with its own structure and numerical demands. Understanding this map explains why the compute substrate is shaped the way it is: the mathematics of each workload dictates whether it is stochastic or deterministic, explicit or implicit, cheap to parallelize or communication-bound.
The families
Kronos solves: the linear Boltzmann transport equation, stochastically, for neutronics; nonlinear elliptic PDEs, the Grad-Shafranov equilibrium; time-dependent PDE systems, resistive MHD and transport; a high-dimensional kinetic equation, gyrokinetics; and coupled decay-transmutation ODE systems for activation. Each is a distinct mathematical object requiring a distinct method.
- Boltzmann transport: stochastic Monte Carlo
- Grad-Shafranov: nonlinear elliptic PDE
- MHD and transport: time-dependent PDE systems
- Gyrokinetics: 5D kinetic PDE
- Activation: coupled linear ODE systems
Method follows structure
The equation's structure chooses the method. High dimensionality and resonance make transport Monte-Carlo-shaped and perfectly parallel. Ellipticity makes equilibrium a boundary-value problem needing a global solve. Stiffness in MHD forces implicit time integration. Coupling between physics forces the multi-physics strategies. The numerics are not arbitrary; they are consequences of the math.
This map also explains the compute demands. Stochastic and independent workloads weak-scale to huge node counts; implicit coupled workloads are communication-bound and strong-scaling-limited; kinetic workloads are the heaviest of all. The substrate's mix of CPU-dense, low-latency, and GPU-dense partitions is a direct answer to this spread of mathematical shapes.
The same mathematics serves both machines. The breeder and burner differ in geometry and dominant physics, but both are described by these equation families. The equations and methods are shared; the boundary conditions, sources, and parameters make one a spherical tokamak and the other a tandem mirror.