  Quantum simulators were originally proposed for simulating one partial
differential equation (PDE) in particular - Schrodinger's equation. Can quantum
simulators also efficiently simulate other PDEs? While most computational
methods for PDEs - both classical and quantum - are digital (PDEs must be
discretised first), PDEs have continuous degrees of freedom. This suggests that
an analog representation can be more natural. While digital quantum degrees of
freedom are usually described by qubits, the analog or continuous quantum
degrees of freedom can be captured by qumodes. Based on a method called
Schrodingerisation, we show how to directly map D-dimensional linear PDEs onto
a (D+1)-qumode quantum system where analog or continuous-variable Hamiltonian
simulation on D+1 qumodes can be used. This very simple methodology does not
require one to discretise PDEs first, and it is not only applicable to linear
PDEs but also to some nonlinear PDEs and systems of nonlinear ODEs. We show
some examples using this method, including the Liouville equation, heat
equation, Fokker-Planck equation, Black-Scholes equations, wave equation and
Maxwell's equations. We also devise new protocols for linear PDEs with random
coefficients, important in uncertainty quantification, where it is clear how
the analog or continuous-variable framework is most natural. This also raises
the possibility that some PDEs may be simulated directly on analog quantum
systems by using Hamiltonians natural for those quantum systems.
