The next generation of supercomputers will have O(1M) processing cores, be able to exploit fine to coarse grained parallelism, benefit from photonic interconnects and GPUs living directly on the processor die, using 3D technology, or nearby as co-processors. These fast changing technologies will enable unprecedented scientific opportunities if correctly exploited. Obviously, the epoch of the "one man - one code - one method" (on one machine architecture) is over. In order for scientists to fully exploit these machines, concentrate on the science at hand and benefit from the latest numerical techniques will point towards a change on how our solvers are developed. In this talk, I will present ideas on how to potentially achieve this. The latter will be ranging from computer science (expression templates, metaprogramming, streaming SIMD execution, load balancing in time-space) to new (and old but under-rated) numerical methods:
adjoint driven AMR using h-p discontinuous Galerkin - PDE - solver, preconditioning and locally (any order) implicit time-integrators.
Numerical simulations presenting the transformative potential of each idea will be shown.
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