Merge pull request #391 from firedrakeproject/Numericalintegrator

PR #391: add numerical integrator

gusto/__init__.py

@@ -13,6 +13,7 @@
 from gusto.limiters import *                         # noqa
 from gusto.linear_solvers import *                   # noqa
 from gusto.meshes import *                           # noqa
+from gusto.numerical_integrator import *             # noqa
 from gusto.physics import *                          # noqa
 from gusto.preconditioners import *                  # noqa
 from gusto.recovery import *                         # noqa

gusto/numerical_integrator.py

@@ -0,0 +1,63 @@
+import numpy as np
+
+
+class NumericalIntegral(object):
+    """
+    A class for numerically evaluating and tabulating some 1D integral.
+    Args:
+        lower_bound(float): lower bound of integral
+        upper_bound(float): upper bound of integral
+        num_points(float): number of points to tabulate integral at
+
+    """
+    def __init__(self, lower_bound, upper_bound, num_points=500):
+
+        # if upper_bound <= lower_bound:
+        #     raise ValueError('lower_bound must be lower than upper_bound')
+        self.x = np.linspace(lower_bound, upper_bound, num_points)
+        self.x_double = np.linspace(lower_bound, upper_bound, 2*num_points-1)
+        self.lower_bound = lower_bound
+        self.upper_bound = upper_bound
+        self.num_points = num_points
+        self.tabulated = False
+
+    def tabulate(self, expression):
+        """
+        Tabulate some integral expression using Simpson's rule.
+        Args:
+            expression (func): a function representing the integrand to be
+                evaluated. should take a numpy array as an argument.
+        """
+
+        self.cumulative = np.zeros_like(self.x)
+        self.interval_areas = np.zeros(len(self.x)-1)
+        # Evaluate expression in advance to make use of numpy optimisation
+        # We evaluate at the tabulation points and the midpoints of the intervals
+        f = expression(self.x_double)
+
+        # Just do Simpson's rule for evaluating area of each interval
+        self.interval_areas = ((self.x[1:] - self.x[:-1]) / 6.0
+                               * (f[2::2] + 4.0 * f[1::2] + f[:-1:2]))
+
+        # Add the interval areas together to create cumulative integral
+        for i in range(self.num_points - 1):
+            self.cumulative[i+1] = self.cumulative[i] + self.interval_areas[i]
+
+        self.tabulated = True
+
+    def evaluate_at(self, points):
+        """
+        Evaluates the integral at some point using linear interpolation.
+        Args:
+            points (float or iter) the point value, or array of point values to
+                evaluate the integral at.
+        Return:
+            returns the numerical approximation of the integral from lower
+            bound to point(s)
+        """
+        # Do linear interpolation from tabulated values
+        if not self.tabulated:
+            raise RuntimeError(
+                'Integral must be tabulated before we can evaluate it at a point')
+
+        return np.interp(points, self.x, self.cumulative)

unit-tests/test_numerical_integrator.py

@@ -0,0 +1,34 @@
+"""
+Tests the numerical integrator.
+"""
+from gusto import NumericalIntegral
+from numpy import sin, pi
+import pytest
+
+
+def quadratic(x):
+    return x**2
+
+
+def sine(x):
+    return sin(x)
+
+
+@pytest.mark.parametrize("integrand_name", ["quadratic", "sine"])
+def test_numerical_integrator(integrand_name):
+    if integrand_name == "quadratic":
+        integrand = quadratic
+        upperbound = 3
+        answer = 9
+    elif integrand_name == "sine":
+        integrand = sine
+        upperbound = pi
+        answer = 2
+    else:
+        raise ValueError(f'{integrand_name} integrand not recognised')
+    numerical_integral = NumericalIntegral(0, upperbound)
+    numerical_integral.tabulate(integrand)
+    area = numerical_integral.evaluate_at(upperbound)
+    err_tol = 1e-10
+    assert abs(area-answer) < err_tol, \
+        f'numerical integrator is incorrect for {integrand_name} function'