Geology 727
MT3DMS Solution Methods and Parameter Options
Introduction
Please refer to Table 3.1 on p. 3-20 of the manual (also attached) for a summary of the solution
methods available in MT3DMS. Note that the there are two main columns in the table. One
column refers to methods used to solve the troublesome advection component of the advectiondispersion equation and the other column refers to solution methods for the rest of the equation
(dispersion, sink/source, and chemical reaction terms). It is useful to think in terms of solving the
advection component and thus to think about four basic solution approaches:
1. MOC/MMOC/HMOC (with either explicit or implicit finite differences for the rest of
the equation);
2. Explicit finite difference method;
3. Implicit finite difference method;
4. TVD method (with either explicit or implicit finite differences for the rest of the
equation).
Time step. The time step is automatically selected by the code to meet various stability
constraints that are solution dependent. By setting the transport time step size (DT0) equal to
zero, the code will automatically select a time step. You may select a different value for the time
step by entering a non-zero value. You must, however, select a value less than the value selected
by the code or else it will default to the code-selected value.
Courant Number. The Courant number (Cr) is used in all solution methods as a way to ensure
that the time step selected by MT3DMS is appropriate for the problem. If the Courant number is
set equal to one, the time step will equal the grid spacing divided by the velocity. This means
that the solution will be calculated for the time it takes for the solute to move across one cell.
MT3DMS selects the time step as follows:
 t = R Cr min (x/vx, y/vy, z/vz)
The Courant number is typically set to a value between 0.5 and 1.0. Sometimes a value up to two
is used. But for the explicit finite difference option and the TVD method, the Courant number is
also a stability constraint and must not exceed one.
Peclet Number. The Peclet number (Pe) measures the ratio of advection to dispersion. Thus, an
advection dominant problem has a high Peclet number.
Pe = v x / D
or
Pe = x / 
Lower order finite difference schemes in MT3DMS (i.e., all finite difference schemes except for
the TVD method) show numerical errors (known as numerical dispersion) for advection
dominant problems. Hence, these methods should be used only when the Peclet number is small,
usually less than four. This can be severely limiting since it limits the magnitude of the
permissible grid spacing.
MOC and HMOC solution methods and the TVD method are not subject to numerical dispersion
and can be used for any value of Peclet number.
MOC/MMOC/HMOC Methods
These methods can be used to solve the advection component of the advection-dispersion
equation. MOC starts the simulation with a fixed number of particles per cell. MMOC uses one
particle per cell. HMOC uses a combination of MOC and MMOC; MOC is used near sharp
concentration fronts and MMOC is used away from such fronts. The MMOC technique should
be used only in situations where sharp concentrations fronts are not present.
In the MOC/MMOC/HMOC methods either the explicit or the implicit finite difference method
is used to solve the dispersion term, sink/source term, and the reaction term. If the explicit finite
difference method is used, various stability constraints arise which set limits on the time step.
See p. 4-20 of the manual for details.
Parameters in the MOC/MMOC/HMOC methods
WD is the concentration weighting factor, which ranges between 0 and 1. The default value is
0.5. This can be adjusted to get better mass balance. Increase toward 1.0 for advection
dominant problems.
NPH is the number of particles per cell. Use a maximum of 32 for 3D problems. Can reduce to
16 for 2D problems.
Number of particles refers to the maximum particles allowed. The default value is 5000. If you
keep introducing particles (solute) into the model domain without removing particles, the
solution may run out of particles, i.e., all 5000 particles will be in use in the model domain and
there won’t be any particles left to continue the introduction of solute. This can create numerical
problems.
Particle Planes refer to the particle placement pattern at the beginning of the simulation. If set to
0, a random pattern is selected. If a value greater than zero is selected, a fixed pattern is used.
See p. 6-28 of the manual for details. The parameter name is NPLANE.
ITRACK selects the particle tracking scheme. Options are a simple 1st order Euler scheme, the
more accurate 4th order Runge Kutta scheme, or a combination such that Runge Kutta is used
near sink and source cells and Euler is used everywhere else.
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The small concentration gradient (DCEPS) is the gradient below which advective transport is
considered negligible, usually set around 1E-5.
DCHMOC is the critical relative concentration gradient for controlling the selective use of either
MOC or MMOC in the HMOC solution. The default value is 0.005.
Finite Difference Methods
Summary
The explicit method is influenced by numerical dispersion and has many restrictions on the time
step. The implicit method is subject to numerical dispersion when the advection term is solved
with upstream weighting; alternatively the method is subject to oscillations when the advection
term is solved with central weighting. It has no stability limit on the time step. Numerical errors
in both methods can be minimized by keeping the Peclet number small (less than four). The
TVD method uses an explicit method to solve the advection term and thus has a stability
constraint on the time step but it is relatively free from numerical errors and can be used for all
values of the Peclet number.
Methods
The explicit method uses upstream weighting for the advection term since the central weighting
scheme is unstable for this method. The explicit method has stability contraints on the time step:
1) the Courant number must be one or less; 2) see p 4-20 of the manual for additional stability
constraints.
The implicit method uses either upstream weighting or central weighting for the advection term.
The time step is controlled by a time step multiplier TSMULT. To use a constant value for the
time step, set the multiplier equal to one. The generalized conjugate gradient (GCG) solver is
used in this method. There is a head change closure criterion on the solver, for which the default
value is 1E-6. There is a choice of three iterative algorithms: Jacobi, SSOR, and Modified
Incomplete Cholesky (MIC). MIC usually takes fewer iterations to converge but requires more
memory.
The explicit 3rd order TVD method uses an explicit finite difference approximation for the
advection term. Therefore, the Courant number must be less than or equal to one. The method
works well for all values of Peclet number. Either standard explicit finite differences or implicit
finite differences are selected to solve the dispersion, sink/source, and chemical reaction terms. If
explicit finite differences are selected, there are additional constraints on the time step. See p. 420 of the manual for details.
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Table 3.1 from MT3DMS manual
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