### Automatic derivation of finite difference stencils

parent cdab4128
fd/derivation.py 0 → 100644
 import sympy as sp from collections import defaultdict from pystencils.field import Field from pystencils.sympyextensions import prod, multidimensional_sum from pystencils.utils import fully_contains, LinearEquationSystem class FiniteDifferenceStencilDerivation: """Derives finite difference stencils. Can derive standard finite difference stencils, as well as isotropic versions (see Isotropic Finite Differences by A. Kumar) Args: derivative_coordinates: tuple indicating which derivative should be approximated, (1, ) stands for first derivative in second direction (y), (0, 1) would be a mixed second derivative in x and y (0, 0, 0) would be a third derivative in x direction stencil: list of offset tuples, defining the stencil dx: spacing between grid points, one for all directions, i.e. dx=dy=dz Examples: Central differences >>> fd_1d = FiniteDifferenceStencilDerivation((0,), stencil=[(-1,), (0,), (1,)]) >>> result = fd_1d.get_stencil() >>> result Finite difference stencil of accuracy 2, isotropic error: False >>> result.weights [-1/2, 0, 1/2] Forward differences >>> fd_1d = FiniteDifferenceStencilDerivation((0,), stencil=[(0,), (1,)]) >>> result = fd_1d.get_stencil() >>> result Finite difference stencil of accuracy 1, isotropic error: False >>> result.weights [-1, 1] """ def __init__(self, derivative_coordinates, stencil, dx=1): self.dim = len(stencil) self.field = Field.create_generic('f', spatial_dimensions=self.dim) self._derivative = tuple(sorted(derivative_coordinates)) self._stencil = stencil self._dx = dx self.weights = {tuple(d): self.symbolic_weight(*d) for d in self._stencil} def assume_symmetric(self, dim, anti_symmetric=False): """Adds restriction that weight in opposite directions of a dimension are equal (symmetric) or the negative of each other (anti symmetric) For example: dim=1, assumes that w(1, 1) == w(1, -1), if anti_symmetric=False or w(1, 1) == -w(1, -1) if anti_symmetric=True """ update = {} for direction, value in self.weights.items(): inv_direction = tuple(-offset if i == dim else offset for i, offset in enumerate(direction)) if direction[dim] < 0: inv_weight = self.weights[inv_direction] update[direction] = -inv_weight if anti_symmetric else inv_weight self.weights.update(update) def set_weight(self, offset, value): assert offset in self.weights self.weights[offset] = value def get_stencil(self, isotropic=False) -> 'FiniteDifferenceStencilDerivation.Result': weights = [self.weights[d] for d in self._stencil] system = LinearEquationSystem(sp.Matrix(weights).atoms(sp.Symbol)) order = 0 while True: new_system = system.copy() eq = self.error_term_equations(order) new_system.add_equations(eq) sol_structure = new_system.solution_structure() if sol_structure == 'single': system = new_system elif sol_structure == 'multiple': system = new_system elif sol_structure == 'none': break else: assert False order += 1 accuracy = order - len(self._derivative) error_is_isotropic = False if isotropic: new_system = system.copy() new_system.add_equations(self.isotropy_equations(order)) sol_structure = new_system.solution_structure() error_is_isotropic = sol_structure != 'none' if error_is_isotropic: system = new_system solve_res = system.solution() weight_list = [self.weights[d].subs(solve_res) for d in self._stencil] return self.Result(self._stencil, weight_list, accuracy, error_is_isotropic) @staticmethod def symbolic_weight(*args): str_args = [str(e) for e in args] return sp.Symbol("w_({})".format(",".join(str_args))) def error_term_dict(self, order): error_terms = defaultdict(lambda: 0) for direction in self._stencil: weight = self.weights[tuple(direction)] x = tuple(self._dx * d_i for d_i in direction) for offset in multidimensional_sum(order, dim=self.field.spatial_dimensions): fac = sp.factorial(order) error_terms[tuple(sorted(offset))] += weight / fac * prod(x[off] for off in offset) if self._derivative in error_terms: error_terms[self._derivative] -= 1 return error_terms def error_term_equations(self, order): return list(self.error_term_dict(order).values()) def isotropy_equations(self, order): def cycle_int_sequence(sequence, modulus): import numpy as np result = [] arr = np.array(sequence, dtype=int) while True: if tuple(arr) in result: break result.append(tuple(arr)) arr = (arr + 1) % modulus return tuple(set(tuple(sorted(t)) for t in result)) error_dict = self.error_term_dict(order) eqs = [] for derivative_tuple in list(error_dict.keys()): if fully_contains(self._derivative, derivative_tuple): remaining = list(derivative_tuple) for e in self._derivative: del remaining[remaining.index(e)] permutations = cycle_int_sequence(remaining, self.dim) if len(permutations) == 1: eqs.append(error_dict[derivative_tuple]) else: for i in range(1, len(permutations)): new_eq = (error_dict[tuple(sorted(permutations[i] + self._derivative))] - error_dict[tuple(sorted(permutations[i - 1] + self._derivative))]) if new_eq: eqs.append(new_eq) else: eqs.append(error_dict[derivative_tuple]) return eqs class Result: def __init__(self, stencil, weights, accuracy, is_isotropic): self.stencil = stencil self.weights = weights self.accuracy = accuracy self.is_isotropic = is_isotropic def visualize(self): from pystencils.stencils import visualize_stencil visualize_stencil(self.stencil, data=self.weights) def apply(self, field_access: Field.Access): f = field_access return sum(f.get_shifted(*offset) * weight for offset, weight in zip(self.stencil, self.weights)) def as_matrix(self): dim = len(self.stencil) assert dim == 2 max_offset = max(max(abs(e) for e in direction) for direction in self.stencil) result = sp.Matrix(2 * max_offset + 1, 2 * max_offset + 1, lambda i, j: 0) for direction, weight in zip(self.stencil, self.weights): result[max_offset - direction, max_offset + direction] = weight return result def __repr__(self): return "Finite difference stencil of accuracy {}, isotropic error: {}".format(self.accuracy, self.is_isotropic)
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