We propose a new paradigm for designing efcient p-adaptive arbitrary high-order meth-ods.We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match...We propose a new paradigm for designing efcient p-adaptive arbitrary high-order meth-ods.We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specifc iteration with the discretization accuracy of the same iteration.Apart from the computational advan-tage,the newly modifed methods allow to naturally perform the p-adaptivity,stopping the iterations when appropriate conditions are met.Moreover,the modifcation 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 Par-tial Diferential Equations(PDEs)is presented here.We explain how such a 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 efcient formula-tion,in which a local a posteriori limiter can be naturally integrated leading to the p-adap-tivity 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 efciency.展开更多
文摘We propose a new paradigm for designing efcient p-adaptive arbitrary high-order meth-ods.We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specifc iteration with the discretization accuracy of the same iteration.Apart from the computational advan-tage,the newly modifed methods allow to naturally perform the p-adaptivity,stopping the iterations when appropriate conditions are met.Moreover,the modifcation 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 Par-tial Diferential Equations(PDEs)is presented here.We explain how such a 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 efcient formula-tion,in which a local a posteriori limiter can be naturally integrated leading to the p-adap-tivity 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 efciency.
