Matlab code to compute the Gram matrices required for Fourier Continuation with the FC-Gram algorithm.
The matrices enable efficient extension of non-periodic functions to periodic ones for spectral differentiation.
FCGram_Matrices(4, 25, 12, 25, 20, 2, 256)| Parameter | Type | Description | Typical Values |
|---|---|---|---|
d |
int | Number of matching points (degree of approximation) | 2-10 (faster convergence for higher d) |
C |
int | Number of continuation points (must be even) | ~25 |
Z |
int | Number of zero padding points for smooth extension | ~C/2 |
E |
int | Number of extra points for numerical stability | ~C |
n_over |
int | Oversampling factor for fine grid construction | >10 |
modes_to_reduce |
int | Number of modes to reduce in SVD truncation | 0-2 |
num_digits |
int | Number of digits for symbolic precision in computations | ≥128, recommended 256 |
d: Typically varies between 2 and 10, with faster convergence for higher valuesC: Seemed to work well with values around 25Z: Works well when roughly equal to C/2E: Works well when roughly equal to Cn_over: Should be greater than 10 for optimal resultsmodes_to_reduce: Can be set to 0 (no impact) or 2 (removes smallest singular values for better approximation)num_digits: Should use at least 128, or 256 for higher precision
ArQr: Right boundary continuation matrix (C × d)AlQl: Left boundary continuation matrix (C × d)
- Construct monomial basis on coarse grid
- Orthonormalize using QR decomposition with full re-orthogonalization
- Evaluate basis on fine grid with oversampling
- Build trigonometric basis for FC approximation
- Compute SVD and solve for coefficients
- Evaluate FC matrix at continuation points
- Construct boundary continuation matrices with flip operations
The function automatically saves the computed matrices to a .mat file in the format:
FCGram_data_d{d}_C{C}.mat
[1] Amlani, F., & Bruno, O. P. (2016). An FC-based spectral solver for elastodynamic problems in general three-dimensional domains. Journal of Computational Physics, 307, 333-354.
- Daniel Leibovici - dleibovi@caltech.edu
- Valentin Duruisseaux - vduruiss@caltech.edu