The elliptic solver allows other types of equations to be more
easily implemented. In this section, we show how the parabolic equation
can be reduced to solving elliptic equations. This is done using the
Partial Differential Equation Toolbox™ solves equations of the form
When the m coefficient is 0, but d is not, the documentation refers to the equation as parabolic, whether or not it is mathematically in parabolic form.
A parabolic problem is to solve the equation
with the initial condition
u(x,0) = u0(x) for x∊Ω
where x represents a 2-D or 3-D point and there are boundary conditions of the same kind as for the elliptic equation on ∂Ω.
The heat equation reads
in the presence of distributed heat loss to the surroundings. ρ is the density, C is the thermal capacity, k is the thermal conductivity, h is the film coefficient, u∞ is the ambient temperature, and f is the heat source.
For time-independent coefficients, the steady-state solution of the equation is the solution to the standard elliptic equation
–∇ · (c∇u) + au = f.
Assuming a mesh on Ω and t ≥ 0, expand the solution to the PDE (as a function of x) in the Finite Element Method basis:
Plugging the expansion into the PDE, multiplying with a test function ϕj, integrating over Ω, and applying Green's formula and the boundary conditions yield
In matrix notation, we have to solve the linear, large and sparse ODE system
This method is traditionally called method of lines semidiscretization.
Solving the ODE with the initial value
Ui(0) = u0(xi)
yields the solution to the PDE at each node xi and time t. Note that K and F are the stiffness matrix and the right-hand side of the elliptic problem
–∇ · (c∇u) + au = f in Ω
with the original boundary conditions, while M is just the mass matrix of the problem
–∇ · (0∇u) + du = 0 in Ω.
When the Dirichlet conditions are time dependent, F contains contributions from time derivatives of h and r. These derivatives are evaluated by finite differences of the user-specified data.
The ODE system is ill conditioned. Explicit time integrators
are forced by stability requirements to very short time steps while
implicit solvers can be expensive since they solve an elliptic problem
at every time step. The numerical integration of the ODE system is
performed by the MATLAB® ODE Suite functions, which are efficient
for this class of problems. The time step is controlled to satisfy
a tolerance on the error, and factorizations of coefficient matrices
are performed only when necessary. When coefficients are time dependent,
the necessity of reevaluating and refactorizing the matrices each
time step may still make the solution time consuming, although
only that which varies with time. In certain cases a time-dependent
Dirichlet matrix h(t)
may cause the error control to fail, even if the problem is mathematically
sound and the solution u(t)
is smooth. This can happen because the ODE integrator looks only at
the reduced solution v with u = Bv + ud.
As h changes, the pivoting scheme
employed for numerical stability may change the elimination order
from one step to the next. This means that B, v,
and ud all change discontinuously, although u itself