# Solve a PDE BVP with Spectral methods in time and space?

I have a PDE (coming from a Bellman equation with $z$ under brownian motion). Let $z \in [0,\infty)$ and $t \in [0,T]$. To sketch the equation: $$(r - g(t)) v(t,z) = \pi(t,z) + (\gamma - g(t))v_z(t,z) + \frac{\sigma^2}{2}v_{zz}(t,z) + v_t(t,z)$$ with a terminal condition $v(T,z)$ and ugly nonlinear boundary conditions involving integrals for $v(t,0)$ and $v_z(t,0)$. There is a very good chance that the PDE will be stiff and $v(t,z)$ fairly close to $v(T,z)$ for all $T$

I am considering two alternatives:

• Use a 2-dimensional Chebyshev basis $$v(t,z)\approx \sum_{i=0}^{N_z}\sum_{j=0}^{N_t}a_{ij}T_i(z)T_j(t)$$ for some $N_z$ and $N_t$ and $T(\cdot)$ are the chebyshev polynomials adapted to the domain. Then solve the BVP using spectral collocation methods for the coefficients (i.e., the tensor product of the polynomial rather than any sparse grid method)
• Use finite-differences in time, and a 1-dimensional chebyshev in space. Hence, for each $t_j$ in the time grid, $$v(t_j,z) \approx \sum_{i=0}^N a^{j}_i T_i(z)$$ and solve for the set of $a^j_i$ for each $j$ in the finite difference scheme, and the set of $N$ nodes at each period.

My question is: If I don't care all that much for extreme precision, but care a great deal about convergence and stability, is either case reasonable or is there a good reason not to use a 2-dimensional Chebyshev basis?

Without going into details, the $\pi(t,z)$ and $g(t)$ come from an outside set of equilibrium conditions that need to be solved, and it would be very convenient to be able to approximate them as chebyshev polynomials themselves (and solve for all of the approximating coefficients for the $g(t)$ and the above $a_{ij}$ at the same time using auto-differentiation and a good nonlinear solver).

(I found this, Spectral Methods in time but the exact set of problems didn't seem clear from the answer. The typical advantages of a finite-difference time stepping are less clear in my case).