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+ Applications include modeling of shocks in plasmas. Finite element method FTol Theo [MP16]. This approach relies on dividing a system/domain into smaller regions called nite elements. The discretization process applied to a boundary value problem leads to a system of algebraic equations, which can be less computationally expensive to resolve than the original PDE. Important applications include uid ow, heat transfer, and electromagnetic potentials. Lattice Boltzmann algorithms Ftol Num [Lju22]. Originally developed as a classical algorithm, this approach can be used to simulate uid dynamics without having to solve the Navier Stokes equations directly. The uid density is represented on a lattice and evolves in time with streaming and collision processes. One of the advantages of the method is its e ciency/scalability in parallel architectures. Quantum lattice algorithms QIns Num [And+22; Kou+22; Oga+18; Ram+21; Vah+20a; Vah+20b; Vah+21a; Vah+21b; Vah+22; VSV20; VVS20; Vah+21c; Vah+20c; Vah+19; Yep02; Yep05; Yep16; VYV03; Vah+11; Vah+10; Oga+16a; Oga+16b; Oga+15; Shi+18]. Highly parallelizable approach amenable to classical supercomputers, allowing the study of (Klein-Gordon-)Maxwell s equations, the Gross-Pitaevski equation, the nonlinear Schr odinger equation, and the KdV equation. In some cases, the method may also be suitable for fault-tolerant quantum computers.