On computer algebra aided numerical solution of ODE by finite … PCA 2019.pdf · 2019. 4. 19. ·...

On computer algebra aided numerical solution of ODEby finite difference method

Yuri A. Blinkov1 Vladimir P. Gerdt2

1Department of Mathematics and MechanicsSaratov University

410012, Saratov, Russia

2Laboratory of Information TechnologiesJoint Institute for Nuclear Research

141980, Dubna, Russia

Talk at PCA 2019, April, St. Petersburg, Russia

Blinkov & Gerd (SSU & JINR) On computer algebra aided numerical solution of ODE PCA 2019 1 / 22

Plan

1 Research problem

2 Numerical results

3 References


Research problem

Except very special cases, nonlinear ordinary differential equations (ODE) admitnumerical integration only. Historically first and one of the most-used numericalmethods is finite difference method (FDM) [1] based on a finite differenceapproximation (FDA). As this takes place, the quality of numerical solution toPDE is determined by the quality of its FDA and by the method of numericalsolution used to solve the difference equations comprising FDA.

One of the most challenging problems in to construct FDA which mimics basicalgebraic properties of the ODE. Such mimetic FDA are more likely to producehighly accurate and stable numerical results (cf. [2]). In particular, a mimetic FDAis to be totally conservative (see [3], Def.1) what means the inheritance ofalgebraic integrals of the ODE at the discrete level.


Research problem

Example. We consider the following autonomous ODE system [3]⎧⎪⎨⎪⎩p =qr ,q = − pr , k = const ,

r = − k2pq

(1)

which has two quadratic integrals

p2 + q2 = const and k2p2 + r2 = const. (2)

We use the denotations x := {p, q, r} and F(x) := {qr ,−pr ,−k2pq} and considerthe implicit midpoint finite difference discretization of system (1)

dxdt

= F(x) =⇒ xtn+1

tn

=

∫ tn+1

tnF(x) dt =⇒ xn+1 − xn

∆t=

F(xn+1) + F(xn)

2(3)


Research problem

The construction of a simple iteration is based on the following quadratic formulasfor monomials occurring in F(x)

u2 = u2 − u′2 + u′2 = (u − u′)(u + u′) +

+ u′2 ≈ (u − u′)2u′ + u′2 = 2uu′ − u′2

uv = (u + v)2 − (u − v)2)/4 ≈ ((2(u + v)(u′ + v ′) − (u′ + v ′)2) −− (2(u − v)(u′ − v ′) − (u′ − v ′)2))/4 = uv ′ + u′v − u′v ′

u3 = u3 − u′3 + u′3 = (u − u′)(u2 + uu′ + u′2) +

+ u′3 ≈ (u − u′)3u′2 + u′2 = 3uu′2 − 2u′3

(4)

and the qubic one

u2v = uuv ≈ u′u′v + uu′v ′ + u′uv ′ −− (3− 1)u′2v ′ = u′2v + 2uu′v ′ − 2u′2v ′ (5)

The rule (5) can be easily implemented in a user’s programming language of anymodern computer algebra system and allows obtain the code for numericalcomputations.


Research problem

Thus, for the system (1) and its implicit scheme (3) the simple ireration methodyields⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

pn+1 − pn

∆t−qn+1r ′n+1 + q′

n+1rn+1 − q′n+1r

′n+1 + qnrn

2= 0,

qn+1 − qn

∆t−−pn+1r ′n+1 − p′

n+1rn+1 + p′n+1r

′n+1 − pnrn

2= 0,

rn+1 − rn∆t

−−k2pn+1q′n+1 − k2p′

n+1qn+1 + k2p′n+1q

′n+1 − k2pnqn

2= 0

(6)


Research problem

In the matrix form obtained by using the computer algebrasystem SymPy (https://www.sympy.org) is given by⎛⎜⎜⎝

1/∆t −r ′n+1/2 −q′n+1/2

r ′n+1/2 1/∆t p′n+1/2

k2q′n+1/2 k2p′

n+1/2 1/∆t

⎞⎟⎟⎠⎛⎝pn+1

qn+1rn+1

⎞⎠ =

=

⎛⎜⎝ pn/∆t − q′n+1r

′n+1/2 + qnrn/2

qn/∆t + p′n+1r

′n+1/2− pnrn/2

rn/∆t + k2p′n+1q

′n+1/2− k2pnqn/2

⎞⎟⎠(7)


https://www.sympy.org

Numerical results

In [3]: p, q, r, k = sympy.symbols(’p, q, r, k’, real=True)

In [4]: f = [q*r, -p*r, -k**2*p*q]f

Out[4]: [qr , −pr , −k2pq

]In [5]: J = sympy.Matrix(f).jacobian(sympy.Matrix([p, q, r]))

J

Out[5]: ⎡⎣ 0 r q−r 0 −p

−k2q −k2p 0

⎤⎦


Numerical results

In [6]: num_f = sympy.lambdify((p, q, r, k), f, "numpy")num_J = sympy.lambdify((p, q, r, k), J, "numpy")

def func(t, y, k):return num_f(y[0], y[1], y[2], k)

def jac(t, y, k):return num_J(y[0], y[1], y[2], k)


Numerical results

lsoda:Real-valued Variable-coefficient Ordinary Differential Equation solver, withfixed-leading-coefficient implementation. It provides automatic method switchingbetween implicit Adams method (for non-stiff problems) and a method based onbackward differentiation formulas (BDF) (for stiff problems).Source: http://www.netlib.org/odepack


http://www.netlib.org/odepack

Numerical results

In [7]: r = ode(func, jac).set_integrator(’lsoda’, method=’bdf’,with_jacobian=True)

r.set_initial_value(y0).set_f_params(num_k,).set_jac_params(num_k,)

sol = ndarray((len(t), 3), dtype=float64)sol[0, :] = y0clock = time.clock()for i in range(1, len(t)):

assert r.successful()r.integrate(t[i])sol[i, :] = r.y

print("Time: %.3f" % (time.clock() - clock))draw(sol)draw(sol, "./fig/fig01.pdf")

Time: 0.283

0 200 400 600 800 1000

t

0.00000

0.00002

0.00004

0.00006

0.00008

p2

+q2

+1 ‖p2 + q2 − 1‖∞ = 8.1e− 05

numericexact


Numerical results

In [7]: r = ode(func, jac).set_integrator(’lsoda’, method=’bdf’,with_jacobian=True)


....Time: 0.283

0 200 400 600 800 1000

t

0.00000

0.00002

0.00004

0.00006

0.00008

p2

+q2

+1 ‖p2 + q2 − 1‖∞ = 8.1e− 05

numericexact


Numerical results

In [8]: r = ode(func).set_integrator(’lsoda’)r.set_initial_value(y0).set_f_params(num_k,)....

Time: 0.366

0 200 400 600 800 1000

t

0.000000

0.000025

0.000050

0.000075

0.000100

0.000125

0.000150

0.000175

p2

+q2

+1 ‖p2 + q2 − 1‖∞ = 1.9e− 04

numericexact


Numerical results

vode:Real-valued Variable-coefficient Ordinary Differential Equation solver, withfixed-leading-coefficient implementation. It provides implicit Adams method (fornon-stiff problems) and a method based on backward differentiation formulas(BDF) (for stiff problems).Authors:

Peter N. Brown and Alan C. HindmarshCenter for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermore, CA 94551

andGeorge D. Byrne (Prof. Emeritus)Illinois Institute of TechnologyChicago, IL 60616

Source: http://www.netlib.org/ode/vode.f


http://www.netlib.org/ode/vode.f

Numerical results

In [9]: r = ode(func, jac).set_integrator(’vode’, method=’bdf’,with_jacobian=True)


....Time: 0.293

0 200 400 600 800 1000

t

1.0000

1.0002

1.0004

1.0006

1.0008

1.0010

1.0012

1.0014

p2

+q2

‖p2 + q2 − 1‖∞ = 1.4e− 03

numericexact


Numerical results

In [10]: r = ode(func).set_integrator(’vode’, method=’bdf’)r.set_initial_value(y0).set_f_params(num_k,)....

Time: 0.338

0 200 400 600 800 1000

t

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

p2

+q2

+1 ‖p2 + q2 − 1‖∞ = 7.3e− 04

numericexact


Numerical results

dopri5, dop853:

This is an explicit runge-kutta method of order (4)5 due to Dormand & Prince(with stepsize control and dense output).

Authors:E. Hairer and G. Wanner Universite de Geneve, Dept. de Mathematiques CH-1211Geneve 24, Switzerland e-mail: [email protected],[email protected]

This code is described in:E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations i.Nonstiff Problems. 2nd edition. Springer Series in Computational Mathematics,Springer-Verlag (1993)


[email protected]

[email protected]

Numerical results

In [12]: r = ode(func).set_integrator(’dopri5’)r.set_initial_value(y0).set_f_params(num_k,)....

Time: 0.740

0 200 400 600 800 1000

t

0.0000065

0.0000070

0.0000075

0.0000080

0.0000085

0.0000090

0.0000095

0.0000100

p2

+q2

+9.9999×10−1 ‖p2 + q2 − 1‖∞ = 3.6e− 06

numericexact


Numerical results

In [13]: r = ode(func).set_integrator(’dop853’)r.set_initial_value(y0).set_f_params(num_k,)....

Time: 1.204

0 200 400 600 800 1000

t

−0.8

−0.6

−0.4

−0.2

0.0

p2

+q2

×10−12 + 1 ‖p2 + q2 − 1‖∞ = 9.2e− 13

numericexact


Numerical results

In [14]: sol = ndarray((len(t), 3), dtype=float64)sol[0, :] = y0tau, num_k2 = t[1] - t[0], num_k**2A = empty((3, 3), dtype=float64)b = empty(3, dtype=float64)y, jmax = y0, 0clock = time.clock()for i in range(1, len(t)):

u, f, j = y, True, 0while f:

A[0] = 1./tau, -u[2]/2., -u[1]/2.A[1] = u[2]/2., 1./tau, u[0]/2.A[2] = num_k2*u[1]/2., num_k2*u[0]/2., 1./taub[:] = y[0]/tau - u[1]*u[2]/2. + y[1]*y[2]/2.,\

y[1]/tau + u[0]*u[2]/2. - y[0]*y[2]/2.,\y[2]/tau + num_k2*u[0]*u[1]/2.

- num_k2*y[0]*y[1]/2.y1 = linalg.solve(A, b)f = norm(y1-u) > 1.e-10u = y1j += 1....Blinkov & Gerd (SSU & JINR) On computer algebra aided numerical solution of ODE PCA 2019 20 / 22

Numerical results

....if j > jmax:

print(j, end= ’ ’)jmax = j

y = y1sol[i, :] = y

print("Time: %.3f" % (time.clock() - clock))draw(sol, start=-100)draw(sol, start=-100, name="./fig/fig08.pdf")

3 4 Time: 0.941

990 992 994 996 998 1000

t

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

p2

+q2

+1 ‖p2 + q2 − 1‖∞ = 6.2e− 04

numericexact


References

References

A. A. Samarskii.Theory of Difference Schemes.Marcel Dekker, New York, 2001.

B. Koren, R. Abgral, P. Bochev, J. Frank and B. Perot (eds.)Physics - compatible numerical methods.J. Comput. Phys., 257, Part B, 1039–1526, 2014.

Edik A. Ayryan, Mikhail D. Malykh, Leonid A. Sevastianov, Yu Ying.Finite Difference Schemes and Classical Transcendental FunctionsLecture Notes in Computer Science, Vol. 11189, pp. 235-242, 2018.

J. M. Sanz-Serna.Symplectic Runge-Kutta schemes for adjoint equations, automaticdifferentiation, optimal control, and more.SIAM Rev. 58(1), 3–33., 2016.


On computer algebra aided numerical solution of ODE by finite … PCA 2019.pdf · 2019. 4. 19. ·...

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