CRT PDE Simulation

Ansys/COMSOL/Dassault. CC0 alternative.

Standard FEM meshes are globally coupled. CRT decomposes boundary conditions into 7 independent channels. Each channel solves a tiny heat equation (mod 8, 9, 25, 49, 11, 13, 17). No inter-channel sync. CRT reconstruct gives the global temperature field. The ring IS the domain decomposition.

How It Works

CRT Domain Decomposition
A 2D heat equation with boundary condition T_bc in Z/214,414,200 decomposes into 7 independent heat equations, one per CRT channel. Channel i solves on Z/m_i with boundary T_bc mod m_i. After convergence, CRT reconstruction gives the global solution. Block-diagonal stiffness matrix: 3,750 entries vs 17,424 coupled (~4.6x reduction). Three data channels (mod 8, 25, 49) carry temperature; four parity channels (mod 9, 11, 13, 17) verify solution integrity.
7 solvers
Zero coupling
Each channel converges independently. No global stiffness matrix.
4.6x fewer ops
Block-diagonal
Coupled: 132^2 = 17,424 entries. CRT: sum(m_i^2) = 3,750 entries.
GPU mapping
7 workgroups
Each PDE solver = 1 GPU workgroup. Zero inter-workgroup sync.
Quad parity
Free check
mod-11, mod-13, and mod-17 channels verify solution integrity. Corruption = instant flag.

Heat Equation Demo

Boundary temp (0-214414199):

Sets boundary condition on 8x8 grid. Runs 4 diffusion steps per channel. CRT reconstructs center temperature.

Heat Map (Canvas)

7 panels: one per CRT channel. 32x32 grid, 4 diffusion steps. Colors: mod 8 (red), mod 9 (green), mod 25 (blue), mod 49 (yellow), mod 11 (magenta), mod 13 (cyan), mod 17 (white).

GPU Parallel Solve (WebGPU)

Real CRT PDE solving on your GPU. 7 frequency bands in WGSL compute shaders.

CRT PDE vs FEM

Stiffness matrixGlobal: N^2 entries, coupledBlock-diagonal: 7 tiny matrices. 3,750 vs 17,424 entries.ParallelismDomain decomposition: sync at boundariesCRT channels: zero sync. Algebraically independent.ConvergenceGlobal iterative (GMRES/CG)Per-channel: tiny systems converge in O(m) steps.VerificationResidual check (expensive)mod-11, mod-13, mod-17 parity: free verification per step.Patent statusAnsys/COMSOL/Dassault patentsCC0. Public domain. Forever.

Source code · Public domain (CC0)

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