Introduction to Numerical Methods in Differential Equations

by Mark H. Holmes

Springer web page

Contact Information

Mark H. Holmes
Department of Mathematical Sciences
Rensselaer Polytechnic Institute, Troy, NY 12180 USA
holmes (@rpi.edu)

Author web page

Page Last Updated: 7/30/2026

Chapter 1: Initial Value Problems

MATLAB Files (directions)

euler.m	Fig 1.3
euler2.m	Fig 1.4
logistic.m	Fig 1.6a
logistic2.m	Fig 1.6a (modified)
leap.m	Fig 1.6b
logistic_error.m	Fig 1.7
energy.m	Fig 1.9a
energy2.m	Fig 1.9b

Videos (Quicktime)

Nanogears 1	Fig 1.1 (Ref: [1])
Nanogears 2	Fig 1.1 (Ref: [1])

Instructor Files

Overheads	for various figures and tables
Animation	showing instability in leapfrog method leap_logistic.m MATLAB code used to produce the video
logistic_error.m	shows error as function of number of grid points (and tol used for trap method)

References

[1]	Nanogears (Fig 1.1) "Molecular dynamics simulations of carbon nanotube-based gears" by J. Han, et al, Nanotechnology 8, 1997, 95-102
[2]	Clothing (Fig 1.5) "A high performance solver for the animation of deformable objects using advanced numerical methods," by Hauth and Etzmuss, Proc. Eurographics 20, 2001, 319-328.
[3]	Planets (Fig 1.8) "Symplectic maps for the n-body problem," by Wisdom and Holman, Astronomical Journal 102, 1991, 1528-1538.

Chapter 2: Boundary Value Problems

MATLAB Files (directions)

example1.m	Fig 2.3
example1s.m	Fig 2.4
example2.m	Fig 2.5
errors.m	Fig 2.6
example4.m	Fig 2.7
example5.m	Fig 2.9
bspline1.m	Fig 2.10a
bspline2.m	Fig 2.10b and comparison with FDM
shooting.m	Fig 2.11
shootings.m	Fig 2.11 but spline interpolation
shooting2.m	shooting for Example 2

Instructor Files

for various figures and tables

References

[1]	for the Birds (Fig 2.1) short film from Pixar

Chapter 3: Diffusion Problems

MATLAB Files (directions)

explicit_sin.m	Fig 3.6
explicit_sin2.m	Fig 3.7
explicit_step.m	Fig 3.8
explicit_error.m	Fig 3.10
implicit_sin.m	Fig 3.12
implicit_step.m	Fig 3.13
crank_sin.m	Fig 3.15
crank_step.m	Fig 3.17
Lrk4_sin.m	Fig 3.19
Lrk4_error.m	Fig 3.20
bspline_error.m	Fig 3.22

Videos (Quicktime)

Fig 3.2	Fig 3.2
Fig 3.3	Fig 3.3
Fig 3.6	N=20, M=5
Fig 3.6	N=20, M=20
Fig 3.6	N=20, M=90
Fig 3.12	N=20, M=5
Fig 3.12	N=20, M=20
Fig 3.15	N=20, M=5
Fig 3.15	N=20, M=20
Fig 3.17	N=20, M=5
Fig 3.17	N=20, M=20

References

[1]	Solidification (Fig 3.1) "Welding: solidification and microstructure," by S. A. David et al, JOM 55, 2003, 14-20.
[2]	Zebra Stripes (Fig 3.1) "Generating textures for arbitrary surfaces using reaction-diffusion," by G. Turk, Computer Graphics (SIGGRAPH '91 Proceedings) 25, 1991, 289-298.
[3]	Vortex transport (Fig 3.21) "Viscous Flow Simulation of a Two-Dimensional Channel Flow with Complex Geometry Using the Grid-Particle Vortex Method," by H. Kudela, ESAIM: Proceedings, Third International Workshop on Vortex Flows and Related Numerical Methods, 1999, 215-224.

Chapter 4: Advection Equation

MATLAB Files (directions)

upwind.m	Fig 4.6 (modified)
downwind.m	Fig 4.6 (modified)
lax.m	Fig 4.11 (modified)

Videos (Quicktime)

Fig 4.2
Fig 4.6	upwind with M=50
Fig 4.6	upwind with M=72
Fig 4.6	upwind with M=100
Fig 4.6	downwind with M=72
Fig 4.10	lax-wendroff with M=70
Fig 4.10	lax-wendroff with M=72
Fig 4.10	lax-wendroff with M=74
Fig 4.10	lax-wendroff with M=100

Instructor Files

for various figures and tables

References

[1]	Vortices (Fig 4.12) "High-Order Numerical Schemes with Positivity-Preserving Properties for the Solution of the Transport Equations in Reactive Flows" by F. M. Denaro, et al, 1995 Meeting of Italian Section of the Combustion Institute, Internet Poster Session.

Chapter 5: Numerical Wave Propagation

MATLAB Files (directions)

wave.m	Fig 5.8
velocities.m	Fig 5.9

Videos (Quicktime)

Fig 5.1
Fig 5.5	k=1.05, w=1.1
Fig 5.5	k=1.1, w=1
Fig 5.5	k=1.1, w=1.05
Fig 5.5	k=1.1, w=1.1
Fig 5.8	M=272
Fig 5.8	M=546
Fig 5.10

References

[1]	Shock Wave (Fig 5.7) "Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow" by W. Henshaw and D. Schwendeman, J. of Comp. Physics, 2007.

Chapter 6: Elliptic Problems

MATLAB Files (directions)

sdm.m	Fig 6.13
cgm.m	Fig 6.14
cgm2.m	Fig 6.15
elaplace.m	Fig 6.16
elaplace2.m	Fig 6.17
plaplace4.f	Fig 6.19 (FORTRAN code)
plaplace5.f	Fig 6.19 (FORTRAN code)
laplace4.f	Fig 6.19 (FORTRAN code)
pclapE5.m	Fig 6.19

Instructor Files

Overheads	for various figures and tables
Animation	showing convergence of SDM (Fig 6.13a)
Animation	showing convergence of SDM (Fig 6.13b) sdm_v.m MATLAB code used to produce the SDM videos
Animation	showing convergence of CGM (Fig 6.14a)
Animation	showing convergence of CGM (Fig 6.14b) cgm_v.m MATLAB code used to produce the CGM videos

Appendix