{
 "schema_version": 1,
 "challenge_id": "string-pluck-v1",
 "submitter": {
  "name": "Jeremy Magland",
  "email": "jmagland@flatironinstitute.org"
 },
 "date": "2026-09-17",
 "viz_url": "https://vault1.magland.org/challenges/string-solutions/site/",
 "submission_id": "fd-second-order",
 "description": "Second-order finite differences: leapfrog in time for the wave and frequency-independent loss terms, with the Kelvin-Voigt term centered in time (implicit, tridiagonal solve per step). Fixed Courant number 0.9, so deliberately dispersive; time step 1/(48000 m). Ordered by increasing oversampling factor m.",
 "code_url": "https://vault1.magland.org/challenges/string-solutions/blob/main/solvers/fd/fd_string.py",
 "solutions": [
  {
   "solution_id": "fd-m1",
   "output": "fd-second-order/fd-m1.wav",
   "method": "finite differences, second order, implicit damping",
   "parameters": {
    "oversampling": 1,
    "n_points": 197,
    "dx_m": 0.003316326530612245,
    "dt_s": 2.0833333333333333e-05,
    "courant": 0.898333
   },
   "timing_seconds": 2.012,
   "hardware": "13th Gen Intel(R) Core(TM) i7-1355U, single thread, Python 3.13.5 + numpy 2.4.4",
   "more_accurate_than": []
  },
  {
   "solution_id": "fd-m2",
   "output": "fd-second-order/fd-m2.wav",
   "method": "finite differences, second order, implicit damping",
   "parameters": {
    "oversampling": 2,
    "n_points": 393,
    "dx_m": 0.0016581632653061226,
    "dt_s": 1.0416666666666666e-05,
    "courant": 0.898333
   },
   "timing_seconds": 4.455,
   "hardware": "13th Gen Intel(R) Core(TM) i7-1355U, single thread, Python 3.13.5 + numpy 2.4.4",
   "more_accurate_than": [
    "fd-m1"
   ]
  },
  {
   "solution_id": "fd-m4",
   "output": "fd-second-order/fd-m4.wav",
   "method": "finite differences, second order, implicit damping",
   "parameters": {
    "oversampling": 4,
    "n_points": 786,
    "dx_m": 0.0008280254777070064,
    "dt_s": 5.208333333333333e-06,
    "courant": 0.899479
   },
   "timing_seconds": 11.354,
   "hardware": "13th Gen Intel(R) Core(TM) i7-1355U, single thread, Python 3.13.5 + numpy 2.4.4",
   "more_accurate_than": [
    "fd-m2"
   ]
  },
  {
   "solution_id": "fd-m8",
   "output": "fd-second-order/fd-m8.wav",
   "method": "finite differences, second order, implicit damping",
   "parameters": {
    "oversampling": 8,
    "n_points": 1571,
    "dx_m": 0.0004140127388535032,
    "dt_s": 2.6041666666666666e-06,
    "courant": 0.899479
   },
   "timing_seconds": 33.202,
   "hardware": "13th Gen Intel(R) Core(TM) i7-1355U, single thread, Python 3.13.5 + numpy 2.4.4",
   "more_accurate_than": [
    "fd-m4"
   ]
  },
  {
   "solution_id": "fd-m16",
   "output": "fd-second-order/fd-m16.wav",
   "method": "finite differences, second order, implicit damping",
   "parameters": {
    "oversampling": 16,
    "n_points": 3142,
    "dx_m": 0.000206940464820121,
    "dt_s": 1.3020833333333333e-06,
    "courant": 0.899766
   },
   "timing_seconds": 111.127,
   "hardware": "13th Gen Intel(R) Core(TM) i7-1355U, single thread, Python 3.13.5 + numpy 2.4.4",
   "fields": {
    "path": "fd-second-order/fd-m16.h5",
    "format": "hdf5"
   },
   "more_accurate_than": [
    "fd-m8"
   ]
  }
 ]
}