Documentation

Coefficients for Systems of PDEs

As Systems of PDEs describes, toolbox functions can address the case of systems of N PDEs. How do you represent the coefficients of your PDE in the correct form? In general, an elliptic system is

(cu)+au=f.

For 2-D systems, the notation (cu) represents an N-by-1 matrix with an (i,1)-component

j=1N(xci,j,1,1x+xci,j,1,2y+yci,j,2,1x+yci,j,2,2y)uj.

For 3-D systems, the notation (cu) represents an N-by-1 matrix with an (i,1)-component

j=1N(xci,j,1,1x+xci,j,1,2y+xci,j,1,3z)uj+j=1N(yci,j,2,1x+yci,j,2,2y+yci,j,2,3z)uj+j=1N(zci,j,3,1x+zci,j,3,2y+zci,j,3,3z)uj.

The symbols a and d denote N-by-N matrices, and f denotes a column vector of length N.

Other problems with N > 1 are the parabolic system

dut(cu)+au=f,

the hyperbolic system

d2ut2(cu)+au=f,

and the eigenvalue system

(cu)+au=λdu.

To solve a PDE using this toolbox, you convert your problem into one of the forms the toolbox accepts. Then express your problem coefficients in a form the toolbox accepts.

The question is how to express each coefficient: d, c, a, and f. For answers, see f Coefficient for Systems, c Coefficient for Systems, and a or d Coefficient for Systems.

    Note:   If any coefficient depends on time or on the solution u or its gradient, then all coefficients should be NaN when either time or the solution u is NaN. This is the way that solvers check to see if the equation depends on time or on the solution.

Was this topic helpful?