In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal e...In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.展开更多
In this paper,the previously proposed second-order process-based modified Patankar Runge-Kutta schemes are extended to the third order of accuracy.Owing to the process-based implicit handling of reactive source terms,...In this paper,the previously proposed second-order process-based modified Patankar Runge-Kutta schemes are extended to the third order of accuracy.Owing to the process-based implicit handling of reactive source terms,the mass conservation,mole balance and energy conservation are kept simultaneously while the positivity for the density and pressure is preserved unconditionally even with stiff reaction networks.It is proved that the first-order truncation terms for the Patankar coefficients must be zero to achieve a prior third order of accuracy for most cases.A twostage Patankar procedure for each Runge-Kutta step is designed to eliminate the first-order truncation terms,accomplish the prior third order of accuracy and maximize the Courant number which the total variational diminishing property requires.With the same approach as the second-order schemes,the third-order ones are applied to Euler equations with chemical reactive source terms.Numerical studies including both 1D and 2D ordinary and partial differential equations are conducted to affirm both the prior order of accuracy and the positivity-preserving property for the density and pressure.展开更多
基金supported by the NSF under Grant DMS-1818467Simons Foundation under Grant 961585.
文摘In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.
基金This work was supported by the National Natural Science Foundation of China(No.12102211)the Science and Technology Innovation 2025 Major Project of Ningbo,China(No.2022Z213).
文摘In this paper,the previously proposed second-order process-based modified Patankar Runge-Kutta schemes are extended to the third order of accuracy.Owing to the process-based implicit handling of reactive source terms,the mass conservation,mole balance and energy conservation are kept simultaneously while the positivity for the density and pressure is preserved unconditionally even with stiff reaction networks.It is proved that the first-order truncation terms for the Patankar coefficients must be zero to achieve a prior third order of accuracy for most cases.A twostage Patankar procedure for each Runge-Kutta step is designed to eliminate the first-order truncation terms,accomplish the prior third order of accuracy and maximize the Courant number which the total variational diminishing property requires.With the same approach as the second-order schemes,the third-order ones are applied to Euler equations with chemical reactive source terms.Numerical studies including both 1D and 2D ordinary and partial differential equations are conducted to affirm both the prior order of accuracy and the positivity-preserving property for the density and pressure.
