The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework i...The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework is the automatic way of getting arbitrarily high order methods,which can be put in the Runge-Kutta(RK)form.The drawback is the larger computational cost with respect to the most used RK methods.To reduce such cost,in an explicit setting,we propose an efcient modifcation:we introduce interpolation processes between the DeC iterations,decreasing the computational cost associated to the low order ones.We provide the Butcher tableaux of the new modifed methods and we study their stability,showing that in some cases the computational advantage does not afect the stability.The fexibility of the novel modifcation allows nontrivial applications to PDEs and construction of adaptive methods.The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.展开更多
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The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework is the automatic way of getting arbitrarily high order methods,which can be put in the Runge-Kutta(RK)form.The drawback is the larger computational cost with respect to the most used RK methods.To reduce such cost,in an explicit setting,we propose an efcient modifcation:we introduce interpolation processes between the DeC iterations,decreasing the computational cost associated to the low order ones.We provide the Butcher tableaux of the new modifed methods and we study their stability,showing that in some cases the computational advantage does not afect the stability.The fexibility of the novel modifcation allows nontrivial applications to PDEs and construction of adaptive methods.The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.
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.
