The transport equation is actually an advection-diffussion-reaction equation, which has the form as
dC/dt + v1dC/dx + v2dC/dx = D(d^2C/dx^2+d^2C/dy^2)-R(x,y)*C+S$$\frac{\partial C}{\partial t} + v_1 \frac{\partial C}{\partial x} + v_2 \frac{\partial C}{\partial x} = D \left(\frac{\partial^2C}{\partial x^2}+\frac{\partial^2C}{\partial y^2}\right)-R(x,y)\cdot C+S$$
where C$C$ is the unknown substrate concentration, v1,v2$v_1$ and $v_2$ are the fluid velocityvelocities in xthe $x$ and y$y$ direction, respectively, D$D$ is the diffusion coefficient, R$R$ is the reaction term, Sand $S$ is the source term. I use the traditional FEM scheme with backward Euler for the time advancing. But it seems that there are some negative values appearing in the numerical solution I solved. Is there any method/scheme to avoid negativeness?
