高分辨率有限差分方法及其應(yīng)用研究
發(fā)布時間：2018-12-15 17:50
【摘要】：許多復(fù)雜的流動現(xiàn)象都可以通過計算流體力學(xué)的手段進行數(shù)值模擬。非線性雙曲型守恒律方程作為流體運動的基本控制方程,對其數(shù)值方法的研究有著重要的科學(xué)意義和應(yīng)用價值。本文主要研究雙曲型守恒律方程的高分辨率有限差分方法,并將其應(yīng)用于各種典型的數(shù)值算例,主要內(nèi)容與研究結(jié)果包括以下幾個方面:1.提出了一種基于WENO重構(gòu)的熵穩(wěn)定格式。以熵穩(wěn)定數(shù)值通量為基礎(chǔ),通過在單元交界面處進行高階WENO重構(gòu),得到一類高分辨率的數(shù)值格式,并用逐維計算的方法將其推廣至二維情形。運用本格式對一維標(biāo)量方程、一維氣體動力學(xué)Euler方程組、一維淺水方程組和二維淺水方程組進行了大量的數(shù)值試驗,并與原熵穩(wěn)定格式的計算結(jié)果比較,結(jié)果表明基于WENO重構(gòu)的熵穩(wěn)定格式能有效提高解在間斷處的分辨率。2.通過引入開關(guān)函數(shù)矩陣,提出了一種求解Euler方程組的自調(diào)節(jié)熵穩(wěn)定格式。開關(guān)函數(shù)具有在光滑區(qū)域接近于0,而在間斷區(qū)域接近于1的性質(zhì)。該函數(shù)自動控制格式在不同位置的數(shù)值耗散大小,使得數(shù)值耗散在間斷區(qū)域自動地添加,從而達到格式的自調(diào)節(jié)性。給出了一維和二維Euler方程組的幾個經(jīng)典算例,驗證了該格式的良好性能。3.提出了一種求解雙曲型守恒律方程的三階熵穩(wěn)定格式。首先基于不同模板上的兩點熵守恒通量的線性組合得到四階熵守恒通量;其次提出一種基于點值的滿足符號性質(zhì)的三階基本無振蕩重構(gòu),利用該重構(gòu)進行(特征)熵變量重構(gòu),設(shè)計了一種三階數(shù)值耗散項,將之添加到四階熵守恒通量,得到了一種三階熵穩(wěn)定格式,并推廣至二維情形。最后通過大量的數(shù)值試驗來檢驗該格式的數(shù)值精度和有效性。數(shù)值結(jié)果表明,該格式在一維和二維情形下均能達到預(yù)期的三階精度,在處理間斷問題時具有高分辨率、基本無振蕩性等優(yōu)點。4.提出了一種求解雙曲型守恒律方程的四階半離散中心迎風(fēng)格式。在Godunov型中心格式的基礎(chǔ)上,充分考慮非線性波的局部傳播速度,利用該速度對Riemann扇的寬度加以精確估計,得到了半離散中心迎風(fēng)數(shù)值通量。將其與Peer的四階基本無振蕩重構(gòu)相結(jié)合,建立了一種四階半離散中心迎風(fēng)格式。該格式無需求解Riemann問題,從而避免了復(fù)雜耗時的特征分解過程。運用該格式求解了標(biāo)量守恒律方程、Euler方程組以及帶坡底源項的淺水方程組。數(shù)值結(jié)果表明,該格式能準(zhǔn)確地計算出解的復(fù)雜細小結(jié)構(gòu),具有高分辨率、基本無振蕩、簡單等優(yōu)良特性。
[Abstract]:Many complex flow phenomena can be numerically simulated by means of computational fluid dynamics. As the basic governing equation of fluid motion, the nonlinear hyperbolic conservation law equation has important scientific significance and application value in the study of its numerical method. In this paper, the high resolution finite difference method for hyperbolic conservation law equations is studied and applied to various typical numerical examples. The main contents and results are as follows: 1. An entropy stable scheme based on WENO reconstruction is proposed. Based on the entropy stable numerical flux, a class of high resolution numerical schemes are obtained by reconstructing high order WENO at the interface of the unit. The scheme is extended to two dimensional cases by using the method of dimensionality calculation. By using this scheme, a large number of numerical experiments have been carried out on one-dimensional scalar equations, one-dimensional gas-dynamics Euler equations, one-dimensional shallow water equations and two-dimensional shallow-water equations, and the results are compared with those of the original entropy stability scheme. The results show that the entropy stable scheme based on WENO reconstruction can effectively improve the resolution at the discontinuity. 2. By introducing the switching function matrix, a self-adjusting entropy stabilization scheme for solving Euler equations is proposed. The switching function is close to 0 in the smooth region and close to 1 in the discontinuous region. This function automatically controls the numerical dissipation size of the scheme at different positions, which makes the numerical dissipation be added automatically in the discontinuous region, thus achieving the self-adjustment of the scheme. Several classical examples of one and two dimensional Euler equations are given, and the good performance of the scheme is verified. A third order entropy stable scheme for solving hyperbolic conservation law equations is proposed. Firstly, based on the linear combination of two-point entropy conservation flux on different templates, the fourth order entropy conservation flux is obtained. Secondly, a third order nonoscillatory reconstruction based on the point value satisfying the symbolic property is proposed. The entropy variable is reconstructed by this reconstruction, and a third order numerical dissipation term is designed, which is added to the fourth order entropy conservation flux. A third order entropy stable scheme is obtained and extended to two dimensional cases. Finally, the numerical accuracy and validity of the scheme are verified by a large number of numerical experiments. The numerical results show that the scheme can achieve the expected third-order accuracy in the case of one and two dimensions, and has the advantages of high resolution and basic non-oscillation in dealing with the discontinuous problems. 4. A fourth order semi-discrete central upwind scheme for solving hyperbolic conservation law equations is proposed. Based on the Godunov central scheme, the local propagation velocity of nonlinear wave is fully considered, and the width of Riemann fan is estimated accurately by using this velocity, and the numerical flux of semi-discrete center upwind is obtained. A fourth order semi-discrete central upwind scheme is established by combining it with the fourth order nonoscillatory reconstruction of Peer. The scheme does not need to solve the Riemann problem, thus avoiding the complex and time-consuming feature decomposition process. By using this scheme, the scalar conservation law equations, Euler equations and shallow water equations with source term at the slope bottom are solved. The numerical results show that the scheme can accurately calculate the complex and fine structure of the solution, and has the advantages of high resolution, basically no oscillation and simplicity.
【學(xué)位授予單位】：西北工業(yè)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O241.82
本文編號：2381053
[Abstract]:Many complex flow phenomena can be numerically simulated by means of computational fluid dynamics. As the basic governing equation of fluid motion, the nonlinear hyperbolic conservation law equation has important scientific significance and application value in the study of its numerical method. In this paper, the high resolution finite difference method for hyperbolic conservation law equations is studied and applied to various typical numerical examples. The main contents and results are as follows: 1. An entropy stable scheme based on WENO reconstruction is proposed. Based on the entropy stable numerical flux, a class of high resolution numerical schemes are obtained by reconstructing high order WENO at the interface of the unit. The scheme is extended to two dimensional cases by using the method of dimensionality calculation. By using this scheme, a large number of numerical experiments have been carried out on one-dimensional scalar equations, one-dimensional gas-dynamics Euler equations, one-dimensional shallow water equations and two-dimensional shallow-water equations, and the results are compared with those of the original entropy stability scheme. The results show that the entropy stable scheme based on WENO reconstruction can effectively improve the resolution at the discontinuity. 2. By introducing the switching function matrix, a self-adjusting entropy stabilization scheme for solving Euler equations is proposed. The switching function is close to 0 in the smooth region and close to 1 in the discontinuous region. This function automatically controls the numerical dissipation size of the scheme at different positions, which makes the numerical dissipation be added automatically in the discontinuous region, thus achieving the self-adjustment of the scheme. Several classical examples of one and two dimensional Euler equations are given, and the good performance of the scheme is verified. A third order entropy stable scheme for solving hyperbolic conservation law equations is proposed. Firstly, based on the linear combination of two-point entropy conservation flux on different templates, the fourth order entropy conservation flux is obtained. Secondly, a third order nonoscillatory reconstruction based on the point value satisfying the symbolic property is proposed. The entropy variable is reconstructed by this reconstruction, and a third order numerical dissipation term is designed, which is added to the fourth order entropy conservation flux. A third order entropy stable scheme is obtained and extended to two dimensional cases. Finally, the numerical accuracy and validity of the scheme are verified by a large number of numerical experiments. The numerical results show that the scheme can achieve the expected third-order accuracy in the case of one and two dimensions, and has the advantages of high resolution and basic non-oscillation in dealing with the discontinuous problems. 4. A fourth order semi-discrete central upwind scheme for solving hyperbolic conservation law equations is proposed. Based on the Godunov central scheme, the local propagation velocity of nonlinear wave is fully considered, and the width of Riemann fan is estimated accurately by using this velocity, and the numerical flux of semi-discrete center upwind is obtained. A fourth order semi-discrete central upwind scheme is established by combining it with the fourth order nonoscillatory reconstruction of Peer. The scheme does not need to solve the Riemann problem, thus avoiding the complex and time-consuming feature decomposition process. By using this scheme, the scalar conservation law equations, Euler equations and shallow water equations with source term at the slope bottom are solved. The numerical results show that the scheme can accurately calculate the complex and fine structure of the solution, and has the advantages of high resolution, basically no oscillation and simplicity.
【學(xué)位授予單位】：西北工業(yè)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2015
【分類號】：O241.82
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