# A Spectral Method for GAMER

## The problem


## Solution through hybrid scheme?

### Run with only one level using the wave scheme

### Run with two levels using the wave scheme

# Goal
- Find a better algorithm to evolve the Schrödinger equation
- Finite Difference, Finite Element, Spectral methods

## Spectral methods
- Expand solutions in terms of a basis of orthogonal functions
- Fourier: trigonometric polynomials
- Chebyshev: Chebyshev polynomials

### Can we use Fourier in each patch? Gibbs phenomenon!


### Can we use a polynomial reconstruction in each patch? Runge phenomenon!


### Chebyshev is the solution?!


## I want to keep using a uniform grid!
- Polynomial subtraction
- Inverse polynomial reconstruction
- Gegenbauer method
- Polynomial least squares
- Fourier-Padé methods
- Filters and Mollifiers
- etc.

## My method of choice: Gram-Fourier extension

### Step 1: Fourier extensions


### Step 2: Approximation at the boundary with orthogonal polynomials


## Step 3: Apply extension to orthogonal polynomials


### The algorithm
1. Expand real and imaginary part in polynomial basis at boundaries
- The boundaries are not the same as GAMER's ghost boundaries
3. Continue polynomials using precomputed periodic extensions
- The information in the extension is unphysical
- We do not use information from neighbouring patches
4. Use FFT on extended domain to evolve wave function
- Wave function is periodic on extended domain
- We largely avoid the Gibb's phenomenon because the extension is very smooth
5. Discard extended domain

### Computational cost
- Small matrix ( 10x10 )
## Problem: Spectral blocking leads to instabilities!


## The restriction mismatch test

### The finite-difference wave scheme


### Finite-difference + Base-level spectral


### Gram-FE + Base-level spectral


## Accuracy tests

### Vortex pair test


#### A nice animation


### Gaussian wave test


#### A nice animation


## Comparison with base-level spectral solver


- Convergence for FFT-based base-level solver with 2 points per wavelength
- Gram FC reaches same accuracy only with roughly 15 points per wavelength
- FD needs many more
- error of 1e-3 for 6 points per wavelength

## Soliton

### Finite-difference scheme with two refinement levels (inbuilt test case)

#### No minimum density



#### With minimum density


### Gram-Fourier extension (with minimum density)



### Gram-Fourier extension only base-level (with minimum density)


## Cosmological simulation
### Hybrid spectral fluid scheme


### Problem: Pure spectral scheme still suffers from artifacts


### Why? Lohner criterion does not enforce 5 - 6 points per wavelength