Automatic derivation of finite difference stencils

fd/derivation.py

+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[0])
+        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[0])
+            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[1], max_offset + direction[0]] = weight
+            return result
+
+        def __repr__(self):
+            return "Finite difference stencil of accuracy {}, isotropic error: {}".format(self.accuracy,
+                                                                                          self.is_isotropic)