We propose a new paradigm for designing efficient p-adaptive arbitrary high order methods. We consider arbitrary high order iterative schemes that gain one order of accuracy at each iteration and we modify them in order to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the new modified methods allow to naturally perform p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such framework can be interpreted as an arbitrary high order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to p-adaptivity and structure preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

翻译：我们提出了一种新范式，用于设计高效的p自适应任意高阶方法。考虑每步迭代提升一阶精度的任意高阶迭代格式，我们对其进行修正，以使特定迭代所达到的精度与该迭代本身的离散精度相匹配。除计算优势外，新修正方法能自然地实现p自适应性，在满足适当条件时停止迭代。此外，该修正极易融入现有任意高阶迭代格式的实现中，且若原方法可实现并行化，则此修正不会破坏并行化的可能性。本文将该方法应用于双曲型偏微分方程的任意导数（ADER）方法。我们阐释了如何通过将其重构为延迟校正方法，使该框架被理解为任意高阶迭代格式，并如何简单修正以获得更高效的公式化表达——在此框架中，局部后验限制器可被自然集成，从而实现p自适应性及结构保持特性。最后，针对可压缩气体动力学的经典基准问题，对新方法进行了全面测试，以展示其鲁棒性与计算效率。