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12/7/2011
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Titus Beu 2011
Titus BeuUniversity “Babes-Bolyai”Department of Theoretical and Computational PhysicsCluj-Napoca, Romania
5. Basics of molecular dynamics5. Basics of molecular dynamics
Titus Beu 2011
Bibliography
Equations of motion for MD simulations
Non-bonded and bonding potentials
Potential derivatives and forces
Van der Waals potentials
Bonded potentials
Force parameters in heterogeneous systems
Time integration methods
Simple application – Oblique throw
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Titus Beu 2011
Leach, A. R., Molecular Modelling. Principles and applications, Second edition (Prentice Hall, 2001) .
Frenkel , D., Smit, B., Understanding Molecular Simulation. From Algorithms to Applications (Academic Press, 2002).
Rapaport, D.C., The Art of Molecular Dynamics Simulation. Second Edition (Cambridge University Press, 2004).
Allen, M. P., Tildesley, D. J., Computer Simulation of Liquids (Clarendon Press, Oxford, 1991).
Titus Beu 2011
Classical MD --- numerical integration of Newton's equations of motion:
ri – position of particle i
Fi – force acting upon particle i
Forces – typically derive from potential functions:
Conservation of total energy:
2
1 22( , , ), 1,2, ,i
i N
dm i N
dt= =
rF r r r… …
1 2 1 2( , , , ) ( , , , )ii N NU= −∇
rF r r r r r r… …
kinE E U= +
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Titus Beu 2011
Simplest potential – pairwise interactions (absence of external forces):
rij = ri − rj distance vector between particles i and j (rij = |rij|)
j > i prevents double counting of particle pairs
Forces on individual particles:
Newton's third law, fji = − fij, reduces calculations
Most time-consuming component of MD – evaluation of forces – effort ~N2
Tractable calculations – forces expressed analytically
Potential cut off – reduction of computational effort: u(rij) = 0 for rij > Rcut
1
( )N N
ij
i j i
U u r= >
=∑∑
N
i ij
j i≠
=∑F f
Titus Beu 2011
Non-bonded potentials – closed-shell weakly-bonded van der Waals systemsEx. inert gases (Ar, Kr,...), hydrogen-bonded systems (water, DNA, proteins,...),...
v(ri) – applied field or effect of container – dropped for periodic bulk systems
3-body and higher order terms are usually neglected
Bonding potentials – strongly bonded systems (covalent bond, ionic bond,...)Ex. hydrocarbons, fullerenes, DNA, proteins, ...
Most of the established force fields sum up harmonic contributions
Force constants weight the deviations from the equilibrium values of the bond
lengths rij, bond angles θijk, etc.
non-bonded( ) ( ) ( , ) ( , , )N
i i j i j k
i i j i i j i k j
U v u w> > >
= + + +∑ ∑∑ ∑∑∑r r r r r r r …
( ) ( ) ( )2 2
0
bonded
bonds angles torsions
( ) 1 cos2 2 2
r n
ij ijk ijklN eq eq
ij ij ijk ijl ijkl ijkl
k k kU r r n
θ
θ θ φ φ = − + − + + − + ∑ ∑ ∑r …
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Titus Beu 2011
AMBER (Assisted Model Building with Energy Refinement) – family of force fieldsfor molecular dynamics of biomolecules
AMBER – software package for MD simulations
Bonded energy between covalently bonded atoms – harmonic approximation
Non-bonded energy between all atom pairs = van der Waals + electrostatic
Reference:D.A. Case, T.A. Darden, T.E. Cheatham, III, C.L. Simmerling, J. Wang, R.E. Duke, R. Luo, R.C. Walker, W. Zhang, K.M. Merz, B. Roberts, B. Wang, S. Hayik, A. Roitberg, G. Seabra, I. Kolossváry, K.F. Wong, F. Paesani, J. Vanicek, J. Liu, X. Wu, S.R. Brozell, T. Steinbrecher, H. Gohlke, Q. Cai, X. Ye, J. Wang, M.-J. Hsieh, G. Cui, D.R. Roe, D.H. Mathews, M.G. Seetin, C. Sagui, V. Babin, T. Luchko, S. Gusarov, A. Kovalenko, and P.A. Kollman (2010), AMBER 11, University of California, San Francisco
( ) ( )
( )
2 2
bonds angles
0
torsions
6 61
0 0
1 1 0
( )2 2
1 cos2
24
r
ij ijkN eq eq
ij ij ijk ijl
n
ijkl
ijkl ijkl
N Nij ij i j
ij
i j i ij ij ij
k kU r r
kn
r r q q
r r r
θ
θ θ
φ φ
επε
−
= = +
= − + −
+ + −
+ − +
∑ ∑
∑
∑∑
r
Titus Beu 2011
Inter-atomic distance in Cartesian coordinates:
Gradient of inter-atomic distance:
Pair-wise forces caused by displacement of atom i:
( ) ( ) ( )
( ) ( ) ( )2 2 2
ij x i j y i j z i j
ij i j i j i j
x x y y z z
r x x y y z z
= − + − + −
= − + − + −
r e e e
ij ij ij ij
x y z
i i i i
ij i j i j i j ij
x y z
i ij ij ij ij
r r r r
x y z
r x x y y z z
r r r r
∂ ∂ ∂ ∂= + +
∂ ∂ ∂ ∂
∂ − − −= + + =
∂
e e er
re e e
r
( ) ( ) ( )
( ) ( ) ( )
ij ij ij ij ij
i
i ij i ij ij
ij ij ij ij ij
j i
j ij j ij ij
u r u r r u r
r r r
u r u r r u r
r r r
∂ ∂ ∂ ∂= − = − = −
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂= − = − = = −
∂ ∂ ∂ ∂
rf
r r
rf f
r r
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Cosine of angle between two bonds (j is the vertex):
Gradient of angle cosine in Cartesian coordinates :
( )( ) ( )( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2
cosi j k j i j k j i j k jij kj
ijk
ij kji j i j i j k j k j k j
x x x x y y y y z x z x
r rx x y y z z x x y y z z
θ− − + − − + − −⋅
= =⋅ − + − + − − + − + −
r r
( ) ( )1cos cos
1cos cos
1cos cos
cos cos cos
k j i j
ijk ijk
i ij kj ij
kj ij
ijk ijk
i ij kj ij
ij kj
ijk ijk
k kj ij kj
ijk ijk ijk
j i k
x x x x
x r r r
r r r
r r r
θ θ
θ θ
θ θ
θ θ θ
− −∂= − ∂
∂= − ∂
∂= − ∂
∂ ∂ ∂= − −
∂ ∂ ∂
r r
r
r r
r
r r r
Titus Beu 2011
Three-body interaction depending on angle between bonds:
Forces on atoms i and k:
Force on vertex atom j:
( , , ) (cos )i j k ijku u θ=r r r
( )j i k= − +f f f
(cos ) (cos ) cos (cos )1cos
cos cos
(cos ) (cos ) cos (cos )1cos
cos c
ijk ijk ijk kj ij ijk
i ijk
i ijk i ij kj ij ijk
ijk ijk ijk ij kj ijk
k ijk
k ijk k kj ij kj
u u u
r r r
u u u
r r r
θ θ θ θθ
θ θ
θ θ θ θθ
θ
∂ ∂ ∂ ∂= − = − = − − ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂= − = − = − − ∂ ∂ ∂ ∂
r rf
r r
r rf
r r os ijkθ
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Titus Beu 2011
Lennard-Jones potential – the most common pair potential for non-bonded (van der Waals) systems (1924):
σ – collision diameter (uLJ(σ) = 0)
ε – well depth (interaction strength)
(σ /rij)12 – strong repulsion at small rij (non-bonded overlap of electronic orbitals)
−(σ /rij)6 – weak attraction at large rij – long-range attractive van der Waals
dispersion forces (dipole-dipole interactions due to fluctuating dipoles)
12 6
LJ
6 6
( ) 4
48 1
2
ij
ij ij
ij
ij
ij ij ij ij
u rr r
r r r r
σ σε
ε σ σ
= −
= − ⋅
rf
Titus Beu 2011
Lennard-Jones potential – the most common pair potential for non-bonded (van der Waals) systems (1924):
req = 21/6σ – minimum (uLJ(req) = −ε)
ε – well depth (interaction strength)
(req /rij)12 – strong repulsion at small rij (non-bonded overlap of electronic orbitals)
−(req /rij)6 – weak attraction at large rij – long-range attractive van der Waals
dispersion forces (dipole-dipole interactions due to fluctuating dipoles)
12 6
LJ
6 6
( ) 2
121
eq eq
ij
ij ij
eq eq ij
ij
ij ij ij ij
r ru r
r r
r r
r r r r
ε
ε
= −
= − ⋅
rf
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//===========================================================================
void ForcesL_0(float epsilon, float sigma, float rcut,float x[], float y[], float z[],
float Fx[], float Fy[], float Fz[], int natm, float &Epot)//---------------------------------------------------------------------------
// Returns forces for the Lennard-Jones 12-6 potential// Energies in eV, forces in eV/Angs, distances in Angs
{float factE, factF, rcut2, sigma2;
float dx, dy, dz, sr2, sr6, fpr, fxi, fyi, fzi, r2;int iatm, jatm;
factE = 4.e0 * epsilon; factF = 48.e0 * epsilon;
sigma2 = sigma * sigma; rcut2 = rcut * rcut;Epot = 0.e0;
for (iatm=1; iatm<=natm; iatm++) Fx[iatm] = Fy[iatm] = Fz[iatm] = 0.e0;
for (iatm=1; iatm<=natm-1; iatm++) {for (jatm=iatm+1; jatm<=natm; jatm++) {
dx = x[iatm] - x[jatm];dy = y[iatm] - y[jatm];
dz = z[iatm] - z[jatm];
r2 = dx*dx + dy*dy + dz*dz;if (r2 < rcut2) {
sr2 = sigma2 / r2;sr6 = sr2 * sr2 * sr2;
fpr = factF * sr6 * (sr6 - 0.5e0) / r2; // f/r
fxi = fpr * dx;fyi = fpr * dy;
fzi = fpr * dz;Fx[iatm] += fxi; Fx[jatm] -= fxi;
Fy[iatm] += fyi; Fy[jatm] -= fyi;Fz[iatm] += fzi; Fz[jatm] -= fzi;
Epot += factE * sr6 * (sr6 - 1.e0);
}}
}}
//===========================================================================
void ForcesL_0(float epsilon, float sigma, float rcut,float x[], float y[], float z[],
float Fx[], float Fy[], float Fz[], int natm, float &Epot)//---------------------------------------------------------------------------
// Returns forces for the Lennard-Jones 12-6 potential// Energies in eV, forces in eV/Angs, distances in Angs
{float factE, factF, rcut2, sigma2;
float dx, dy, dz, sr2, sr6, fpr, fxi, fyi, fzi, r2;int iatm, jatm;
factE = 4.e0 * epsilon; factF = 48.e0 * epsilon;
sigma2 = sigma * sigma; rcut2 = rcut * rcut;Epot = 0.e0;
for (iatm=1; iatm<=natm; iatm++) Fx[iatm] = Fy[iatm] = Fz[iatm] = 0.e0;
for (iatm=1; iatm<=natm-1; iatm++) {for (jatm=iatm+1; jatm<=natm; jatm++) {
dx = x[iatm] - x[jatm];dy = y[iatm] - y[jatm];
dz = z[iatm] - z[jatm];
r2 = dx*dx + dy*dy + dz*dz;if (r2 < rcut2) {
sr2 = sigma2 / r2;sr6 = sr2 * sr2 * sr2;
fpr = factF * sr6 * (sr6 - 0.5e0) / r2; // f/r
fxi = fpr * dx;fyi = fpr * dy;
fzi = fpr * dz;Fx[iatm] += fxi; Fx[jatm] -= fxi;
Fy[iatm] += fyi; Fy[jatm] -= fyi;Fz[iatm] += fzi; Fz[jatm] -= fzi;
Epot += factE * sr6 * (sr6 - 1.e0);
}}
}}
Titus Beu 2011
General Lennard-Jones potential (1924, John Lennard-Jones):
n = 12, m = 6 Lennard-Jones potential
Buckingham exp-6 potential (1938):
r−12 replaced by a more realistic exp
3 adjustable parameters (ε, rm, α)
Negative singularity (strongly attractive) in the origin!
/( )
LJ ( )
n mm n m
ij
ij ij
n nu r
n m m r r
σ σε
− = − −
6
B
6( ) exp 1
6 6
mij
m
rru r
r r
αε α
α α
= − − −
− −
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Titus Beu 2011
Hill potential (1948):
Parameters – determined by fitting to data for the rare gases
Assumed to be applicable to non-polar gases
Halgren potential (1992):
rij* – depends on the atomic polarizability rii* = Aαi1/4
n = 12, m = 6, δ = γ = 0 → Lennard-Jones potential
n = 14, m = 7, δ = 0.07, γ = 0.12 → buffered 14-7 potential
( ) ( )
( )
H * *
1 1( ) 2
/ /
n m
ij
ij ij ij ij
u rr r r r
δ γε
δ γ
− + + = −
+ +
( ) ( )6
Hill ( ) expij ij m m iju r A B r r C r r= − −
Titus Beu 2011
Many-body effects can significantly affect the dispersion interaction – account for approximately 10% of the lattice energy of crystalline argon
Axilrod-Teller three-body dispersion potential (1943) – triple-dipole correction:
Linear arrangement of atoms – negative contribution – enhances interaction(high correlation of the motions of the electrons)
Equilateral arrangement of atoms – positive contribution – weakens interaction(low correlations)
( ) ( )( )
AT 3 3 3
2 2 2
5 5 5
1 3cos cos cos( , , )
3
ijk jki kij
i j k
ij jk ki
ij jk ki ij jk jk ki ki ij
ij jk ki
u Ar r r
r r rA
r r r
θ θ θ+=
+ ⋅ ⋅ ⋅=
r r r
r r r r r r
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Finnis-Sinclair potential (1984) – pairwise contributions + many-body term:
u(rij) – pairwise potential (can include electrostatic and repulsive contributions)
Many-body term ~ electron density (decays exponentially from the nuclei)
Tersoff potential (1988) – “empirical bond-order potential”:
bij – bond order between atom i and atom j – the bond strength decreases as the number of bonds to atom i increases
( ) ( )1/2
( ) ( )
( ) exp exp
N
FS ij i
i j i
N
FS ij ij
i j i j i
U u r B
U A r B r
ρ
α β
>
> ≠
= −
= − − −
∑ ∑
∑ ∑ ∑
r
r
( ) ( )
( )1/2
,
( ) exp exp
1 exp
N
T ij ij ij
i j i j i
ij ik ij
k i k j
U A r B b r
b r r
α β
β
> ≠
−
≠ ≠
= − − −
= + − −
∑ ∑ ∑
∑
r
Titus Beu 2011
The most common potential for bonded interactions – covalent bonds:
k – force constant for bond stretching
req – equilibrium bond length
rcut – cut-off distance
u'bond – shifted bonding potential – asymptotically 0 (discontinuous forces in rcut)
( )
( )
( )
2
bond eq
2
bond eq bond cut
eq
( )2
( ) ( )2
ij ij
ij ij
ij
ij ij
ij
ku r r r
ku r r r u r
k r rr
= −
′ = − −
= − − ⋅r
f
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//===========================================================================
void ForcesB_0(float kfor, float req, float rcut,float x[], float y[], float z[],
float Fx[], float Fy[], float Fz[], int natm, float &Epot)//---------------------------------------------------------------------------
// Returns harmonic forces for bonded interactions// Energies in eV, forces in eV/Angs, distances in Angs
{float factE, factF, Ecut;
float dr, dx, dy, dz, fpr, fxi, fyi, fzi, r;int iatm, jatm;
factE = 0.5e0 * kfor;
factF = -kfor;dr = rcut - req;
Ecut = factE * dr * dr;Epot = 0.e0;
for (iatm=1; iatm<=natm; iatm++) Fx[iatm] = Fy[iatm] = Fz[iatm] = 0.e0;
for (iatm=1; iatm<=natm-1; iatm++) {for (jatm=iatm+1; jatm<=natm; jatm++) {
dx = x[iatm] - x[jatm];dy = y[iatm] - y[jatm];
dz = z[iatm] - z[jatm];
r = sqrt(dx*dx + dy*dy + dz*dz);if (r <= rcut) {
dr = r - req;fpr = factF * dr / r; // f/r
fxi = fpr * dx;fyi = fpr * dy;
fzi = fpr * dz;Fx[iatm] += fxi; Fx[jatm] -= fxi;
Fy[iatm] += fyi; Fy[jatm] -= fyi;Fz[iatm] += fzi; Fz[jatm] -= fzi;
Epot += factE * dr * dr - Ecut;
}}
}}
//===========================================================================
void ForcesB_0(float kfor, float req, float rcut,float x[], float y[], float z[],
float Fx[], float Fy[], float Fz[], int natm, float &Epot)//---------------------------------------------------------------------------
// Returns harmonic forces for bonded interactions// Energies in eV, forces in eV/Angs, distances in Angs
{float factE, factF, Ecut;
float dr, dx, dy, dz, fpr, fxi, fyi, fzi, r;int iatm, jatm;
factE = 0.5e0 * kfor;
factF = -kfor;dr = rcut - req;
Ecut = factE * dr * dr;Epot = 0.e0;
for (iatm=1; iatm<=natm; iatm++) Fx[iatm] = Fy[iatm] = Fz[iatm] = 0.e0;
for (iatm=1; iatm<=natm-1; iatm++) {for (jatm=iatm+1; jatm<=natm; jatm++) {
dx = x[iatm] - x[jatm];dy = y[iatm] - y[jatm];
dz = z[iatm] - z[jatm];
r = sqrt(dx*dx + dy*dy + dz*dz);if (r <= rcut) {
dr = r - req;fpr = factF * dr / r; // f/r
fxi = fpr * dx;fyi = fpr * dy;
fzi = fpr * dz;Fx[iatm] += fxi; Fx[jatm] -= fxi;
Fy[iatm] += fyi; Fy[jatm] -= fyi;Fz[iatm] += fzi; Fz[jatm] -= fzi;
Epot += factE * dr * dr - Ecut;
}}
}}
Titus Beu 2011
Three-body interaction depending harmonically on angle between bonds:
Forces on atoms i and k:
Force on vertex atom j:
( )2
( , , ) (cos ) cos cos2
ijk eq
i j k ijk ijk ijk
ku u
θ
θ θ θ= = −r r r
( )
( )
cos cos(cos )1cos cos
cos
cos cos(cos )1cos c
cos
eq
ijk ijk ijkkj ij ijk kj ij
i ijk ijk
ij kj ij ijk ij kj ij
eq
ijk ijk ijkkj ij ijk ij
k ijk
ij kj ij ijk kj ij
ku
r r r r r r
ku
r r r r r
θ
θ
θ θθθ θ
θ
θ θθθ
θ
− ∂= − − = − − ∂
− ∂= − − = − − ∂
r r r rf
r r rf os
kj
ijk
kjrθ
r
( )j i k= − +f f f
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Titus Beu 2011
Lorentz-Berthelot mixing rules – standard approach for deriving parameters for mixtures from atomic parameters
Lennard-Jones interaction:collision diameter – arithmetic mean, well depth – geometric mean:
Most successful when applied to similar species
Failure for dissimilar species due to overestimation of the well depth
van der Waals parameters derived from:
crystal packing analysis
liquid simulations optimized to reproduce thermodynamic properties (densities, enthalpies of vaporisation,...)
( ) ( )1 1,
2 2
eq eq eq
AB A B AB AA BB
AB AA BB
r r rσ σ σ
ε ε ε
= + = +
=
Titus Beu 2011
Standard method for solving ODEs – finite difference approach
All finite difference integrators assume that trajectories are continuous and the positions and velocities can be represented as Taylor series expansions:
Solution on a step-by-step basis – propagated solution becomes initial condition for next step
δt is significantly smaller (femtoseconds) than the trajectory extent (nanoseconds)
Order of integrator – power of δt in the last term retained in the Taylor series
MD integrators:
Low-order methods – leapfrog, Verlet – easy implementation, stability
High-order (predictor-corrector) methods – high accuracy for large time-steps
( )
( )
2 3 4
2 3
1 1( ) ( ) ( ) ( ) ( )
2 6
1( ) ( ) ( ) ( )
2
t t t t t t t t t O t
t t t t t t t O t
δ δ δ δ δ
δ δ δ δ
+ = + + + +
+ = + + +
r r v a b
v v a b
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Titus Beu 2011
Euler algorithm
Approximation – quadratic in positions and linear in velocities:
Advantage: simplicity
Disadvantages:
lowest order algorithm – O(δt3)
significant total energy drift (propagation of round-off errors)
not time-reversible
Not recommended!
( )
( )
2 3
2
1( ) ( ) ( ) ( )
2
( ) ( ) ( )
t t t t t t t O t
t t t t t O t
δ δ δ δ
δ δ δ
+ = + + +
+ = + +
r r v a
v v a
Titus Beu 2011
Verlet algorithm (Verlet, 1967)
Connects positions at times t − δt, t, and t + δt:
Propagation relation (by summation) – implies only positions and accelerations:
Velocities are not intrinsic quantities – necessary for kinetic energy & temperature:
Advantages: O(δt4) algorithm, small storage requirements, energy conservation
Disadvantages:
propagates round-off errors – difference of large (δt)0 terms + small (δt)2 term
non-synchronized velocities – lag of 1 or ½ step behind positions
not self-starting – besides r(t), also r(t−δt) is needed
( )
( )
2 3 4
2 3 4
1 1( ) ( ) ( ) ( ) ( )
2 6
1 1( ) ( ) ( ) ( ) ( )
2 6
t t t t t t t t t O t
t t t t t t t t t O t
δ δ δ δ δ
δ δ δ δ δ
+ = + + + +
− = − + − +
r r v a b
r r v a b
( )2 4( ) 2 ( ) ( ) ( )t t t t t t t O tδ δ δ δ+ = − − + +r r r a
[ ] ( ) ( )2( ) ( ) ( ) 2t t t t t t O tδ δ δ δ= + − − +v r r
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Titus Beu 2011
Leap Frog algorithm (Hockney & Eastwood, 1970)
Mid-interval velocities:
Propagation relations – subtract and replace first one up to O(δt2) in r(t+δt/2):
Velocities and positions are staggered – velocity at time t:
Advantages: O(δt3) algorithm, small storage requirements, energy conservation
Disadvantages:
non-synchronized velocities – lag of ½ step behind positions
not self-starting – besides r(t), also v(t−δt/2) is needed
( )
( )
2 3
2 3
( 2) ( ) ( ) 2 ( ) 8
( 2) ( ) ( ) 2 ( ) 8
t t t t t t t O t
t t t t t t t O t
δ δ δ δ
δ δ δ δ
+ = + + +
− = − + +
v v a b
v v a b
( )
( )
3
3
( 2) ( 2) ( )
( ) ( ) ( 2)
t t t t t t O t
t t t t t t O t
δ δ δ δ
δ δ δ δ
+ = − + +
+ = + + +
v v a
r r v
[ ] ( )31( ) ( 2) ( 2)
2t t t t t O tδ δ δ= − + + +v v v
Titus Beu 2011
Velocity Verlet algorithm (Swope et al., 1982)
Positions, velocities and accelerations at the same time
Algebraically equivalent to Verlet and Leap Frog algorithms
1. Predictor step – advance velocity to mid-interval and propagate position:
2. Corrector step – update force (acceleration) and propagate velocity:
Advantages:
O(δt3) algorithm
small storage requirements
good energy conservation
( )( ) ( )
( ) ( 2) ( ) 2
t t t t m
t t t t t t t
δ δ
δ δ δ δ
+ = +
+ = + + +
a F r
v v a
( )
( )
2
3
( 2) ( ) ( ) 2
( ) ( ) ( 2)
t t t t t O t
t t t t t t O t
δ δ δ
δ δ δ δ
+ = + +
+ = + + +
v v a
r r v
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Titus Beu 2011
//===========================================================================
void Verlet0(float amass, float fpar, float rpar, float rcut, float dt,float x[], float y[], float z[],
float vx[], float vy[], float vz[],float ax[], float ay[], float az[], int natm,
float &Ekin, float &Epot, int fopt)//---------------------------------------------------------------------------
// Energies in eV, forces in eV/A, distances in A, velocities in A/psec,// accelerations in A/psec^2
//---------------------------------------------------------------------------{
float dt2;int iatm;
dt2 = 0.5e0 * dt;
for (iatm=1; iatm<=natm; iatm++) {vx[iatm] += ax[iatm] * dt2; // v(t+dt/2)
vy[iatm] += ay[iatm] * dt2;vz[iatm] += az[iatm] * dt2;
x[iatm] += vx[iatm] * dt; // r(t+dt)y[iatm] += vy[iatm] * dt;
z[iatm] += vz[iatm] * dt;}
switch(fopt) { // F(t+dt)
case 0: ForcesL_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); break;case 1: ForcesB_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); }
Ekin = 0.e0;
for (iatm=1; iatm<=natm; iatm++) {ax[iatm] /= amass; // a(t+dt)
ay[iatm] /= amass;az[iatm] /= amass;
vx[iatm] += ax[iatm] * dt2; // v(t+dt)vy[iatm] += ay[iatm] * dt2;
vz[iatm] += az[iatm] * dt2;Ekin += vx[iatm]*vx[iatm] + vy[iatm]*vy[iatm] + vz[iatm]*vz[iatm];
}Ekin *= 0.5e0 * amass;
}
//===========================================================================
void Verlet0(float amass, float fpar, float rpar, float rcut, float dt,float x[], float y[], float z[],
float vx[], float vy[], float vz[],float ax[], float ay[], float az[], int natm,
float &Ekin, float &Epot, int fopt)//---------------------------------------------------------------------------
// Energies in eV, forces in eV/A, distances in A, velocities in A/psec,// accelerations in A/psec^2
//---------------------------------------------------------------------------{
float dt2;int iatm;
dt2 = 0.5e0 * dt;
for (iatm=1; iatm<=natm; iatm++) {vx[iatm] += ax[iatm] * dt2; // v(t+dt/2)
vy[iatm] += ay[iatm] * dt2;vz[iatm] += az[iatm] * dt2;
x[iatm] += vx[iatm] * dt; // r(t+dt)y[iatm] += vy[iatm] * dt;
z[iatm] += vz[iatm] * dt;}
switch(fopt) { // F(t+dt)
case 0: ForcesL_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); break;case 1: ForcesB_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); }
Ekin = 0.e0;
for (iatm=1; iatm<=natm; iatm++) {ax[iatm] /= amass; // a(t+dt)
ay[iatm] /= amass;az[iatm] /= amass;
vx[iatm] += ax[iatm] * dt2; // v(t+dt)vy[iatm] += ay[iatm] * dt2;
vz[iatm] += az[iatm] * dt2;Ekin += vx[iatm]*vx[iatm] + vy[iatm]*vy[iatm] + vz[iatm]*vz[iatm];
}Ekin *= 0.5e0 * amass;
}
Titus Beu 2011
Pedictor-corrector methods
Higher order than the Verlet method – require extra computations and storage
Multiple-value methods:
multistep methods – accelerations from several earlier time steps –Adams approach
higher derivatives methods – higher accelerations at current time step –Nordsieck method
We consider multistep methods – stepsize h cannot be changed easily
The method implies two steps:
predictor step providing an initial approximation to the propagated solution
corrector step yielding a refined approximation.
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Predictor step – extrapolation to time t+δt of values from earlier time steps:
Adams-Bashforth method (O(δtk+1))
Provides exact results for x(t) = tq with q ≤ k if the (rational) coefficients αi satisfy the k−1 equations:
[ ]
[ ]
12
1
1
1
( ) : ( ) ( ) ( ) ( ) (1 )
( ) ( )( ) : ( ) (1 )
k
i
i
k
i
i
P x x t t x t v t t t a t i t
x t t x tP v v t t t a t i t
t
δ δ δ α δ
δδ δ α δ
δ
−
=
−
=
+ = + + + −
+ −′+ = + + −
∑
∑
1
1
1
1
1(1 ) ,
( 1)( 2)
1(1 ) , 0,1, , 2
2
kq
i
i
kq
i
i
iq q
i q kq
α
α
−
=
−
=
− =+ +
′− = = −+
∑
∑ …
Titus Beu 2011
Corrector step – uses the predicted values to update f(t+δt):
the predicted values do not appear explicitly in the corrector formulae, except for the evaluation f:
Adams-Moulton method (O(δtk+1))
[ ]
[ ]
12
1
1
1
( ) : ( ) ( ) ( ) ( ) (2 )
( ) ( )( ) : ( ) (2 )
k
i
i
k
i
i
C x x t t x t v t t t a t i t
x t t x tC v v t t t a t i t
t
δ δ δ β δ
δδ δ β δ
δ
−
=
−
=
+ = + + + −
+ −′+ = + + −
∑
∑
1
1
1
1
1(2 ) ,
( 1)( 2)
1(2 ) , 0,1, , 2
2
kq
i
i
kq
i
i
iq q
i q kq
β
β
−
=
−
=
− =+ +
′− = = −+
∑
∑ …
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Gear-type predictor-corrector algorithm (k = 4):
1. Predictor step
2. Update forces
3. Corrector step
[ ]
[ ]
2
( ) ( ) ( ) 19 ( ) 10 ( ) 3 ( 2 )24
( ) ( )( ) 27 ( ) 22 ( ) 7 ( 2 )
24
tt t t t t t t t t t
t t t tt t t t t t t
t
δδ δ δ δ
δ δδ δ δ
δ
+ = + + − − + −
+ −+ = + − − + −
r r v a a a
r rv a a a
[ ]
[ ]
2
( ) ( ) ( ) 3 ( ) 10 ( ) ( )24
( ) ( )( ) 7 ( ) 6 ( ) ( )
24
tt t t t t t t t t t
t t t tt t t t t t t
t
δδ δ δ δ
δ δδ δ δ
δ
+ = + + + + − −
+ −+ = + + + − −
r r v a a a
r rv a a a
( ) ( , ( ), ( )) /t t t t t t t t mδ δ δ δ+ = + + +a F r v
Titus Beu 2011
//===========================================================================
void PredCorr0(float amass, float fpar, float rpar, float rcut, float dt,float x[], float y[], float z[],
float vx[], float vy[], float vz[],float ax[], float ay[], float az[],
float ax1[], float ay1[], float az1[],float ax2[], float ay2[], float az2[], int natm,
float &Ekin, float &Epot, int iopt, int fopt)//---------------------------------------------------------------------------
// x[Angs], v[Angs/psec], a[Angs/psec*psec]{
const float pr0 = 19.e0/24.e0, pr1 =-10.e0/24.e0, pr2 = 3.e0/24.e0;const float pv0 = 27.e0/24.e0, pv1 =-22.e0/24.e0, pv2 = 7.e0/24.e0;
const float cr0 = 3.e0/24.e0, cr1 = 10.e0/24.e0, cr2 =-1.e0/24.e0;const float cv0 = 7.e0/24.e0, cv1 = 6.e0/24.e0, cv2 =-1.e0/24.e0;
float *x0, *y0, *z0, *vx0, *vy0, *vz0;
float fact, dt1, dt2;int iatm;
if (iopt == 0) {
x0 = Vector(1,natm); vx0 = Vector(1,natm);y0 = Vector(1,natm); vy0 = Vector(1,natm);
z0 = Vector(1,natm); vz0 = Vector(1,natm);}
dt2 = dt * dt;
//===========================================================================
void PredCorr0(float amass, float fpar, float rpar, float rcut, float dt,float x[], float y[], float z[],
float vx[], float vy[], float vz[],float ax[], float ay[], float az[],
float ax1[], float ay1[], float az1[],float ax2[], float ay2[], float az2[], int natm,
float &Ekin, float &Epot, int iopt, int fopt)//---------------------------------------------------------------------------
// x[Angs], v[Angs/psec], a[Angs/psec*psec]{
const float pr0 = 19.e0/24.e0, pr1 =-10.e0/24.e0, pr2 = 3.e0/24.e0;const float pv0 = 27.e0/24.e0, pv1 =-22.e0/24.e0, pv2 = 7.e0/24.e0;
const float cr0 = 3.e0/24.e0, cr1 = 10.e0/24.e0, cr2 =-1.e0/24.e0;const float cv0 = 7.e0/24.e0, cv1 = 6.e0/24.e0, cv2 =-1.e0/24.e0;
float *x0, *y0, *z0, *vx0, *vy0, *vz0;
float fact, dt1, dt2;int iatm;
if (iopt == 0) {
x0 = Vector(1,natm); vx0 = Vector(1,natm);y0 = Vector(1,natm); vy0 = Vector(1,natm);
z0 = Vector(1,natm); vz0 = Vector(1,natm);}
dt2 = dt * dt;
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// PREDICTOR
for (iatm=1; iatm<=natm; iatm++) {x0[iatm] = x[iatm]; vx0[iatm] = vx[iatm];
y0[iatm] = y[iatm]; vy0[iatm] = vy[iatm];z0[iatm] = z[iatm]; vz0[iatm] = vz[iatm];
x[iatm] = x0[iatm] + vx0[iatm] * dt
+ (pr0*ax[iatm] + pr1*ax1[iatm] + pr2*ax2[iatm]) * dt2;y[iatm] = y0[iatm] + vy0[iatm] * dt
+ (pr0*ay[iatm] + pr1*ay1[iatm] + pr2*ay2[iatm]) * dt2;z[iatm] = z0[iatm] + vz0[iatm] * dt
+ (pr0*az[iatm] + pr1*az1[iatm] + pr2*az2[iatm]) * dt2;
vx[iatm] = (x[iatm] - x0[iatm]) / dt+ (pv0*ax[iatm] + pv1*ax1[iatm] + pv2*ax2[iatm]) * dt;
vy[iatm] = (y[iatm] - y0[iatm]) / dt+ (pv0*ay[iatm] + pv1*ay1[iatm] + pv2*ay2[iatm]) * dt;
vz[iatm] = (z[iatm] - z0[iatm]) / dt+ (pv0*az[iatm] + pv1*az1[iatm] + pv2*az2[iatm]) * dt;
ax2[iatm] = ax1[iatm]; ax1[iatm] = ax[iatm];
ay2[iatm] = ay1[iatm]; ay1[iatm] = ay[iatm];az2[iatm] = az1[iatm]; az1[iatm] = az[iatm];
}
switch(fopt) { // F(t+dt)case 0: ForcesL_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); break;
case 1: ForcesB_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); }
// PREDICTOR
for (iatm=1; iatm<=natm; iatm++) {x0[iatm] = x[iatm]; vx0[iatm] = vx[iatm];
y0[iatm] = y[iatm]; vy0[iatm] = vy[iatm];z0[iatm] = z[iatm]; vz0[iatm] = vz[iatm];
x[iatm] = x0[iatm] + vx0[iatm] * dt
+ (pr0*ax[iatm] + pr1*ax1[iatm] + pr2*ax2[iatm]) * dt2;y[iatm] = y0[iatm] + vy0[iatm] * dt
+ (pr0*ay[iatm] + pr1*ay1[iatm] + pr2*ay2[iatm]) * dt2;z[iatm] = z0[iatm] + vz0[iatm] * dt
+ (pr0*az[iatm] + pr1*az1[iatm] + pr2*az2[iatm]) * dt2;
vx[iatm] = (x[iatm] - x0[iatm]) / dt+ (pv0*ax[iatm] + pv1*ax1[iatm] + pv2*ax2[iatm]) * dt;
vy[iatm] = (y[iatm] - y0[iatm]) / dt+ (pv0*ay[iatm] + pv1*ay1[iatm] + pv2*ay2[iatm]) * dt;
vz[iatm] = (z[iatm] - z0[iatm]) / dt+ (pv0*az[iatm] + pv1*az1[iatm] + pv2*az2[iatm]) * dt;
ax2[iatm] = ax1[iatm]; ax1[iatm] = ax[iatm];
ay2[iatm] = ay1[iatm]; ay1[iatm] = ay[iatm];az2[iatm] = az1[iatm]; az1[iatm] = az[iatm];
}
switch(fopt) { // F(t+dt)case 0: ForcesL_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); break;
case 1: ForcesB_0(fpar,rpar,rcut,x,y,z,ax,ay,az,natm,Epot); }
Titus Beu 2011
Ekin = 0.e0;
for (iatm=1; iatm<=natm; iatm++) {ax[iatm] /= amass; // a(t+dt)
ay[iatm] /= amass;az[iatm] /= amass;
x[iatm] = x0[iatm] + vx0[iatm] * dt
+ (cr0*ax[iatm] + cr1*ax1[iatm] + cr2*ax2[iatm]) * dt2;y[iatm] = y0[iatm] + vy0[iatm] * dt
+ (cr0*ay[iatm] + cr1*ay1[iatm] + cr2*ay2[iatm]) * dt2;z[iatm] = z0[iatm] + vz0[iatm] * dt
+ (cr0*az[iatm] + cr1*az1[iatm] + cr2*az2[iatm]) * dt2;
vx[iatm] = (x[iatm] - x0[iatm]) / dt+ (cv0*ax[iatm] + cv1*ax1[iatm] + cv2*ax2[iatm]) * dt;
vy[iatm] = (y[iatm] - y0[iatm]) / dt+ (cv0*ay[iatm] + cv1*ay1[iatm] + cv2*ay2[iatm]) * dt;
vz[iatm] = (z[iatm] - z0[iatm]) / dt+ (cv0*az[iatm] + cv1*az1[iatm] + cv2*az2[iatm]) * dt;
Ekin += vx[iatm]*vx[iatm] + vy[iatm]*vy[iatm] + vz[iatm]*vz[iatm];
}Ekin *= 0.5e0 * amass;
if (iopt < 0) {
FreeVector(x0,1);FreeVector(y0,1);
FreeVector(z0,1);}
}
Ekin = 0.e0;
for (iatm=1; iatm<=natm; iatm++) {ax[iatm] /= amass; // a(t+dt)
ay[iatm] /= amass;az[iatm] /= amass;
x[iatm] = x0[iatm] + vx0[iatm] * dt
+ (cr0*ax[iatm] + cr1*ax1[iatm] + cr2*ax2[iatm]) * dt2;y[iatm] = y0[iatm] + vy0[iatm] * dt
+ (cr0*ay[iatm] + cr1*ay1[iatm] + cr2*ay2[iatm]) * dt2;z[iatm] = z0[iatm] + vz0[iatm] * dt
+ (cr0*az[iatm] + cr1*az1[iatm] + cr2*az2[iatm]) * dt2;
vx[iatm] = (x[iatm] - x0[iatm]) / dt+ (cv0*ax[iatm] + cv1*ax1[iatm] + cv2*ax2[iatm]) * dt;
vy[iatm] = (y[iatm] - y0[iatm]) / dt+ (cv0*ay[iatm] + cv1*ay1[iatm] + cv2*ay2[iatm]) * dt;
vz[iatm] = (z[iatm] - z0[iatm]) / dt+ (cv0*az[iatm] + cv1*az1[iatm] + cv2*az2[iatm]) * dt;
Ekin += vx[iatm]*vx[iatm] + vy[iatm]*vy[iatm] + vz[iatm]*vz[iatm];
}Ekin *= 0.5e0 * amass;
if (iopt < 0) {
FreeVector(x0,1);FreeVector(y0,1);
FreeVector(z0,1);}
}
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#include <math.h>
#include <stdio.h>#include <conio.h>
#include "Memalloc.h"#include <Graphlib.h>
//===========================================================================
void Forces(float m, float x, float y, float z, float vx, float vy, float vz,float &Fx, float &Fy, float &Fz, float &Epot)
//---------------------------------------------------------------------------{
const float g = 9.81;const float k = 1.0;
Fx = - k*vx;
Fy = - k*vy;Fz = -m*g - k*vz;
Epot = m * g * z;
}
#include <math.h>
#include <stdio.h>#include <conio.h>
#include "Memalloc.h"#include <Graphlib.h>
//===========================================================================
void Forces(float m, float x, float y, float z, float vx, float vy, float vz,float &Fx, float &Fy, float &Fz, float &Epot)
//---------------------------------------------------------------------------{
const float g = 9.81;const float k = 1.0;
Fx = - k*vx;
Fy = - k*vy;Fz = -m*g - k*vz;
Epot = m * g * z;
}
Titus Beu 2011
//===========================================================================
void Propag(float m, float &x, float &y, float &z,float &vx, float &vy, float &vz,
float &ax, float &ay, float &az,float &Ekin, float &Epot, float dt)
//---------------------------------------------------------------------------{
float dt2;
dt2 = 0.5 * dt;vx += ax * dt2; x += vx * dt;
vy += ay * dt2; y += vy * dt;vz += az * dt2; z += vz * dt;
Forces(m,x,y,z,vx,vy,vz,ax,ay,az,Epot);
ax /= m; ay /= m; az /= m;
vx += ax * dt2;vy += ay * dt2;
vz += az * dt2;
Ekin = 0.5*m*(vx*vx + vy*vy + vz*vz); }
//===========================================================================
void Propag(float m, float &x, float &y, float &z,float &vx, float &vy, float &vz,
float &ax, float &ay, float &az,float &Ekin, float &Epot, float dt)
//---------------------------------------------------------------------------{
float dt2;
dt2 = 0.5 * dt;vx += ax * dt2; x += vx * dt;
vy += ay * dt2; y += vy * dt;vz += az * dt2; z += vz * dt;
Forces(m,x,y,z,vx,vy,vz,ax,ay,az,Epot);
ax /= m; ay /= m; az /= m;
vx += ax * dt2;vy += ay * dt2;
vz += az * dt2;
Ekin = 0.5*m*(vx*vx + vy*vy + vz*vz); }
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Titus Beu 2011
//===========================================================================
void Display(float x[],float y[],float z[],float vx[],float vy[],float vz[],float Ekin[],float Epot[],float Etot[],int i)
//---------------------------------------------------------------------------{
int icont;float *step;
cleardevice(); step = Vector(1,i);
for(icont=1; icont <= i; icont++) {step[icont]=icont;}
Plot(step,x,i,1, 0.1,0.4,0.7,0.9,"istep","x","");Plot(step,y,i,1, 0.1,0.4,0.4,0.6,"istep","y","");
Plot(step,z,i,1, 0.1,0.4,0.1,0.3,"istep","z","");Plot(step,Ekin,i,1, 0.5,0.9,0.7,0.9,"istep","Ekin","");
Plot(step,Epot,i,1, 0.5,0.9,0.4,0.6,"istep","Epot","");Plot(step,Etot,i,1, 0.5,0.9,0.1,0.3,"istep","Etot","");
FreeVector(step,1); }
//===========================================================================
void Display(float x[],float y[],float z[],float vx[],float vy[],float vz[],float Ekin[],float Epot[],float Etot[],int i)
//---------------------------------------------------------------------------{
int icont;float *step;
cleardevice(); step = Vector(1,i);
for(icont=1; icont <= i; icont++) {step[icont]=icont;}
Plot(step,x,i,1, 0.1,0.4,0.7,0.9,"istep","x","");Plot(step,y,i,1, 0.1,0.4,0.4,0.6,"istep","y","");
Plot(step,z,i,1, 0.1,0.4,0.1,0.3,"istep","z","");Plot(step,Ekin,i,1, 0.5,0.9,0.7,0.9,"istep","Ekin","");
Plot(step,Epot,i,1, 0.5,0.9,0.4,0.6,"istep","Epot","");Plot(step,Etot,i,1, 0.5,0.9,0.1,0.3,"istep","Etot","");
FreeVector(step,1); }
Titus Beu 2011
//===========================================================================
int main(){
const float pi = 3.1415926;int maxstep = 10000;
float *x,*vx,*ax;
float *y,*vy,*ay;float *z,*vz,*az;
float *Ekin,*Epot,*Etot;float Ekini,Epoti,Etoti;
float m,xi,yi,zi,vxi,vyi,vzi,axi,ayi,azi;float v,alfa,beta,dt;
float x0,y0,z0,vx0,vy0,vz0;int i,n;
// xmax =
// tmax = // nstep = tmax / dt;
x = Vector(1,maxstep); vx = Vector(1,maxstep); ax = Vector(1,maxstep);
y = Vector(1,maxstep); vy = Vector(1,maxstep); ay = Vector(1,maxstep);z = Vector(1,maxstep); vz = Vector(1,maxstep); az = Vector(1,maxstep);
Ekin = Vector(1,maxstep);
Epot = Vector(1,maxstep);Etot = Vector(1,maxstep);
//===========================================================================
int main(){
const float pi = 3.1415926;int maxstep = 10000;
float *x,*vx,*ax;
float *y,*vy,*ay;float *z,*vz,*az;
float *Ekin,*Epot,*Etot;float Ekini,Epoti,Etoti;
float m,xi,yi,zi,vxi,vyi,vzi,axi,ayi,azi;float v,alfa,beta,dt;
float x0,y0,z0,vx0,vy0,vz0;int i,n;
// xmax =
// tmax = // nstep = tmax / dt;
x = Vector(1,maxstep); vx = Vector(1,maxstep); ax = Vector(1,maxstep);
y = Vector(1,maxstep); vy = Vector(1,maxstep); ay = Vector(1,maxstep);z = Vector(1,maxstep); vz = Vector(1,maxstep); az = Vector(1,maxstep);
Ekin = Vector(1,maxstep);
Epot = Vector(1,maxstep);Etot = Vector(1,maxstep);
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// Inits
m = 1.0;x0 = 0.0;
y0 = 0.0;z0 = 10.0;
v = 10;alfa = 0.0; alfa = alfa/180.0 * pi;
beta = 30.0; beta = beta/180.0 * pi;vx0 = v * sin(beta) * cos(alfa);
vy0 = v * sin(beta) * sin(alfa);vz0 = v * cos(beta);
dt = 0.01;
initwindow(1000,750);
i = 1;
xi = x0; vxi = vx0;yi = y0; vyi = vy0;
zi = z0; vzi = vz0;Forces(m,xi,yi,zi,vxi,vyi,vzi,axi,ayi,azi,Epoti);
x[1] = xi; vx[1] = vxi; ax[1] = axi/m;
y[1] = yi; vy[1] = vyi; ay[1] = ayi/m;z[1] = zi; vz[1] = vzi; az[1] = azi/m;
Ekin[1] = 0.5*m*(vxi*vxi + vyi*vyi + vzi*vzi);Epot[1] = Epoti;
Etot[1] = Ekini + Epoti;
// Inits
m = 1.0;x0 = 0.0;
y0 = 0.0;z0 = 10.0;
v = 10;alfa = 0.0; alfa = alfa/180.0 * pi;
beta = 30.0; beta = beta/180.0 * pi;vx0 = v * sin(beta) * cos(alfa);
vy0 = v * sin(beta) * sin(alfa);vz0 = v * cos(beta);
dt = 0.01;
initwindow(1000,750);
i = 1;
xi = x0; vxi = vx0;yi = y0; vyi = vy0;
zi = z0; vzi = vz0;Forces(m,xi,yi,zi,vxi,vyi,vzi,axi,ayi,azi,Epoti);
x[1] = xi; vx[1] = vxi; ax[1] = axi/m;
y[1] = yi; vy[1] = vyi; ay[1] = ayi/m;z[1] = zi; vz[1] = vzi; az[1] = azi/m;
Ekin[1] = 0.5*m*(vxi*vxi + vyi*vyi + vzi*vzi);Epot[1] = Epoti;
Etot[1] = Ekini + Epoti;
Titus Beu 2011
while (zi >= 0.0) {
Propag(m,xi,yi,zi,vxi,vyi,vzi,axi,ayi,azi,Ekini,Epoti,dt);printf("%f %f %f\n",xi,yi,zi);
i++;x[i] = xi; vx[i] = vxi; ax[i] = axi/m;
y[i] = yi; vy[i] = vyi; ay[i] = ayi/m;z[i] = zi; vz[i] = vzi; az[i] = azi/m;
Ekin[i] = Ekini;Epot[i] = Epoti;
Etot[i] = Ekini + Epoti;
Display(x,y,z, vx,vy,vz, Ekin,Epot,Etot,i);}
n = i;
Display(x,y,z, vx,vy,vz, Ekin,Epot,Etot,n);getch();
closegraph();
FreeVector(x,1); FreeVector(vx,1); FreeVector(ax,1);FreeVector(y,1); FreeVector(vy,1); FreeVector(ay,1);
FreeVector(z,1); FreeVector(vz,1); FreeVector(az,1);FreeVector(Ekin,1);
FreeVector(Epot,1);FreeVector(Etot,1);
return 0;}
while (zi >= 0.0) {
Propag(m,xi,yi,zi,vxi,vyi,vzi,axi,ayi,azi,Ekini,Epoti,dt);printf("%f %f %f\n",xi,yi,zi);
i++;x[i] = xi; vx[i] = vxi; ax[i] = axi/m;
y[i] = yi; vy[i] = vyi; ay[i] = ayi/m;z[i] = zi; vz[i] = vzi; az[i] = azi/m;
Ekin[i] = Ekini;Epot[i] = Epoti;
Etot[i] = Ekini + Epoti;
Display(x,y,z, vx,vy,vz, Ekin,Epot,Etot,i);}
n = i;
Display(x,y,z, vx,vy,vz, Ekin,Epot,Etot,n);getch();
closegraph();
FreeVector(x,1); FreeVector(vx,1); FreeVector(ax,1);FreeVector(y,1); FreeVector(vy,1); FreeVector(ay,1);
FreeVector(z,1); FreeVector(vz,1); FreeVector(az,1);FreeVector(Ekin,1);
FreeVector(Epot,1);FreeVector(Etot,1);
return 0;}
12/7/2011
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Titus Beu 2011
