In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19...In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimen- sional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function inte- gral is approximated by applying standard one dimensional high order numerical quadra- tures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the in this paper achieve or exceed the expected second to fourth-order methods implemented accuracy.展开更多
Flows containing steady or nearly steady strong shocks on parts of the flow field,and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing t...Flows containing steady or nearly steady strong shocks on parts of the flow field,and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing the same numerical scheme,even under the multiblock grid or adaptive grid refinement framework.While sixthorder or higher-order shock-capturing methods are appropriate for unsteady turbulence with shocklets,third-order or lower shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence.In order to minimize the short comings of low order and high order shock-capturing schemes for the subject flows,a multiblock overlapping grid with different types of spatial schemes and orders of accuracy on different blocks is proposed.The recently developed single block high order filter scheme in generalized geometries for Navier Stokes and magnetohydrodynamics systems is extended to multiblock overlapping grid geometries.The first stage in validating the high order overlapping approach with several test cases is included.展开更多
Three high order shock-capturing schemes are compared for large eddy simulations(LES)of temporally evolving mixing layers for different convective Mach numbers ranging from the quasi-incompressible regime to highly co...Three high order shock-capturing schemes are compared for large eddy simulations(LES)of temporally evolving mixing layers for different convective Mach numbers ranging from the quasi-incompressible regime to highly compressible supersonic regime.The considered high order schemes are fifth-order WENO(WENO5),seventh-order WENO(WENO7)and the associated eighth-order central spatial base scheme with the dissipative portion of WENO7 as a nonlinear post-processing filter step(WENO7fi).This high order nonlinear filter method of Yee&Sjogreen is designed for accurate and efficient simulations of shock-free compressible turbulence,turbulence with shocklets and turbulence with strong shocks with minimum tuning of scheme parameters.The LES results by WENO7fi using the same scheme parameter agree well with experimental results compiled by Barone et al.,and published direct numerical simulations(DNS)work of Rogers&Moser and Pantano&Sarkar,whereas results by WENO5 andWENO7 compare poorly with experimental data and DNS computations.展开更多
We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations.We focus on...We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations.We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations.Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space.For time stability,small corrections near the ends of the interval are computed using local polynomial interpolation.Several numerical experiments illustrate the performance of the method.展开更多
文摘In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimen- sional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function inte- gral is approximated by applying standard one dimensional high order numerical quadra- tures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the in this paper achieve or exceed the expected second to fourth-order methods implemented accuracy.
基金This work performed under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344。
文摘Flows containing steady or nearly steady strong shocks on parts of the flow field,and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing the same numerical scheme,even under the multiblock grid or adaptive grid refinement framework.While sixthorder or higher-order shock-capturing methods are appropriate for unsteady turbulence with shocklets,third-order or lower shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence.In order to minimize the short comings of low order and high order shock-capturing schemes for the subject flows,a multiblock overlapping grid with different types of spatial schemes and orders of accuracy on different blocks is proposed.The recently developed single block high order filter scheme in generalized geometries for Navier Stokes and magnetohydrodynamics systems is extended to multiblock overlapping grid geometries.The first stage in validating the high order overlapping approach with several test cases is included.
基金Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
文摘Three high order shock-capturing schemes are compared for large eddy simulations(LES)of temporally evolving mixing layers for different convective Mach numbers ranging from the quasi-incompressible regime to highly compressible supersonic regime.The considered high order schemes are fifth-order WENO(WENO5),seventh-order WENO(WENO7)and the associated eighth-order central spatial base scheme with the dissipative portion of WENO7 as a nonlinear post-processing filter step(WENO7fi).This high order nonlinear filter method of Yee&Sjogreen is designed for accurate and efficient simulations of shock-free compressible turbulence,turbulence with shocklets and turbulence with strong shocks with minimum tuning of scheme parameters.The LES results by WENO7fi using the same scheme parameter agree well with experimental results compiled by Barone et al.,and published direct numerical simulations(DNS)work of Rogers&Moser and Pantano&Sarkar,whereas results by WENO5 andWENO7 compare poorly with experimental data and DNS computations.
文摘We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations.We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations.Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space.For time stability,small corrections near the ends of the interval are computed using local polynomial interpolation.Several numerical experiments illustrate the performance of the method.
