dev.nm.analysis.differentialequation.ode.ivp.solver

Class EulerMethod

java.lang.Object

dev.nm.analysis.differentialequation.ode.ivp.solver.rungekutta.RungeKutta

dev.nm.analysis.differentialequation.ode.ivp.solver.EulerMethod

All Implemented Interfaces:

ODESolver

public class EulerMethod
extends RungeKutta

The Euler method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge-Kutta method. The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. It also suffers from stability problems. For these reasons, the Euler method is not often used in practice. It serves as the basis to construct more complicated methods.

Choosing a grid

The integer constructor specifies the number of grid points, including both endpoints: new EulerMethod(6) uses five intervals. The double constructor specifies a step size: new EulerMethod(0.2). For a forward interval supply a positive step size. Read the returned x() values instead of assuming an exactly uniform grid, because the last grid point is replaced with the requested endpoint.

Euler is first order and can be unstable. Check convergence by reducing the step size, or choose an appropriate higher-order solver.

Example

 import dev.nm.algebra.linear.vector.doubles.Vector;
 import dev.nm.algebra.linear.vector.doubles.dense.DenseVector;
 import dev.nm.analysis.differentialequation.ode.ivp.problem.DerivativeFunction;
 import dev.nm.analysis.differentialequation.ode.ivp.problem.ODE1stOrder;
 import dev.nm.analysis.differentialequation.ode.ivp.solver.EulerMethod;
 import dev.nm.analysis.differentialequation.ode.ivp.solver.ODESolution;

 DerivativeFunction derivative = new DerivativeFunction() {
     public int dimension() { return 1; }
     public Vector evaluate(double t, Vector y) {
         return new DenseVector(new double[]{-y.get(1)});
     }
 };
 ODE1stOrder problem = new ODE1stOrder(derivative,
         new DenseVector(new double[]{1.0}), 0.0, 1.0);
 ODESolution solution = new EulerMethod(6).solve(problem);
 assert solution.x().length == 6; // arrays use zero-based indexing
 assert solution.x()[0] == 0.0;
 assert solution.x()[5] == 1.0;
 assert Math.abs(solution.y()[5].get(1) - Math.pow(0.8, 5)) < 1e-12;
 

See Also:

• D. Greenspan, Numerical Solution of Ordinary Differential Equations: for Classical, Relativistic and Nano Systems, 1st ed, Wiley-VCH, 2006.
• Wikipedia: Euler method

Constructor Summary

Constructors

Constructor and Description
EulerMethod(double h) Constructs an Euler's method instance, with the given step size.
EulerMethod(int N) Constructs an Euler's method instance, with the given number of grid points, including both endpoints.

Method Summary

Methods inherited from class dev.nm.analysis.differentialequation.ode.ivp.solver.rungekutta.RungeKutta

solve

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

EulerMethod

public EulerMethod(double h)

Constructs an Euler's method instance, with the given step size.

Parameters:

h - the constant step size

EulerMethod

public EulerMethod(int N)

Constructs an Euler's method instance, with the given number of grid points, including both endpoints.

Parameters:

N - the number of grid points, including both endpoints; at least 2