Summary: We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundary-value problems for systems of PDEs. We prove computability of the solution operator for a symmetric hyperbolic system with computable real coefficients and dissipative boundary conditions, and of the Cauchy problem for the same system (we also prove computable dependence on the coefficients) in a cube \(Q\subseteq\mathbb R^m\). Such systems describe a wide variety of physical processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many boundary-value problems for the wave equation also can be reduced to this case, thus we partially answer a question raised in [K. Weihrauch and N. Zhong, Proc. Lond. Math. Soc. (3) 85, No. 2, 312–332 (2002; Zbl 1011.03035)]. Compared with most of other existing methods of proving computability for PDEs, this method does not require existence of explicit solution formulas and is thus applicable to a broader class of (systems of) equations.