I don't have an exact answer to your question (if it actually exists); but I can answer part of your question concerned with the I/O to a quantum processor.
As a general rule of thumb; Quantum Algorithms (currently) cannot provide direct answers to problem statements. At least for now, quantum processors exists as heterogeneous accelerators with a classical computing unit. The 'quantum accelerator' is concerned with only that part of the overall algorithm that is not trivial (or exponential in complexity) to solve on a classical computer. In the end, only a sub portion of the program is actually computed on the quantum processor. (Eg. Shor's Factoring Algorithm is actually a period finding algorithm. Period finding is a non-trivial task.)
Among several other reasons, of the main problems is input and output operation with a quantum processor. The problem 'must' be expressible in a concise form (eg. an equation). This equation is expressed as a quantum circuit in the 'oracle' which is primarily concerned with solving the equation and measurement outcome are recorded (tomography). The output too needs post processing to actually make sense (which is again performed by the classical counterpart).
p.s. I would be very interested to know more about PDE solving quantum algorithms; if there is an efficient one.