Bicompact schemes are generalized for the first time to the linear multidimensional convection–diffusion equation. Schemes are constructed using the method of lines, the finite-volume method, and bi- and tricubic Hermite interpolation of the sought function in a cell. Time stepping is based on diagonally implicit Runge–Kutta methods. The proposed bicompact schemes are unconditionally stable, conservative, and fourth-order accurate in space for sufficiently smooth solutions. The constructed schemes are implemented by applying an efficient iterative method based on approximate factorization of their multidimensional equations. Every iteration of the method is reduced to a set of independent one-dimensional scalar two-point Gaussian eliminations. Several stationary and nonstationary exact solutions are used to demonstrate the high-order convergence of the developed schemes and the fast convergence of their iterative implementation. Advantages of bicompact schemes as compared with Galerkin-type finite-element schemes are discussed.

For the first time, a method is proposed for constructing a multidimensional combined shock-capturing scheme that monotonically localizes shock wave fronts and, at the same time, has increased accuracy in smoothness regions of calculated generalized solutions. In this method, the solution of the combined scheme is constructed using monotonic solutions of a bicompact scheme of the first order of approximation in time and fourth order of approximation in space. The solutions are obtained for different time steps in the entire computational domain. This construction method is much simpler than other techniques for constructing combined schemes with similar properties. The results of test calculations are presented, which demonstrate the high accuracy of the proposed combined scheme as applied to a multidimensional flow with shock waves.

For the unsteady incompressible Navier–Stokes equations, a high-order accurate bicompact scheme having the fourth order of approximation in space and the second order of approximation in time has been constructed for the first time. The scheme is obtained by applying the Marchuk–Strang splitting method with respect to physical processes. The convective part of the equations is discretized by additionally using locally one-dimensional splitting. The grid convergence of the proposed scheme with an order higher than the theoretical one is demonstrated on the exact solution of the two-dimensional Taylor–Green vortex problem. The developed bicompact scheme is used to compute the decay of the three-dimensional Taylor–Green vortex (in both laminar and turbulent regimes). It is shown that the scheme well resolves vortex structures and reproduces the turbulent spectrum of kinetic energy with high accuracy.

Bicompact schemes that have the fourth order of classical approximation in space and a higher order (at least the second) in time are considered. Their accuracy is studied as applied to a quasilinear hyperbolic system of conservation laws with discontinuous solutions involving shock waves with variable propagation velocities. The shallow water equations are used as an example of such a system. It is shown that a nonmonotone bicompact scheme has a higher order of convergence in domains of influence of unsteady shock waves. If spurious oscillations are suppressed by applying a conservative limiting procedure, then the bicompact scheme, though being high-order accurate on smooth solutions, has a reduced (first) order of convergence in the domains of influence of shock waves.

In this work, a new limiting method for bicompact schemes is proposed that preserves them conservative. The method is based upon a finite-element treatment of the bicompact approximation. An analogy between collocation finite-element schemes and bicompact schemes is established. The proposed method is tested on one-dimensional gas dynamics problems that include the Sedov problem, the “peak test” Riemann problem, the Shu–Osher problem, and the “blast wave” problem. Additionally, the method is tested as applied to a two-dimensional problem for the quasilinear Hopf equation. It is shown on these examples that bicompact schemes with conservative limiting are significantly more accurate than hybrid bicompact schemes.

О р д е н а Л е н и н а ИНСТИТУТ ПРИКЛАДНОЙ МАТЕМАТИКИ имени М.В.Келдыша Р о с с и й с к о й а к а д е м и и н а у к Б.В.Рогов Бикомпактная интерполяционнохарактеристическая схема третьего порядка аппроксимации

A new method is proposed for constructing a combined shock-capturing scheme that monotonically localizes shock wave fronts and, at the same time, has increased accuracy in smoothness regions of calculated generalized solutions. In this method, the solution of the combined scheme is constructed using monotonic solutions of a bicompact scheme of the first order of approximation in time and the fourth order of approximation in space obtained for different time steps in the entire computational domain. This construction method is much simpler than a previously proposed method. Test calculations are presented that demonstrate the advantages of the new scheme compared to the WENO5 scheme of the fifth order of approximation in space and the third order of approximation in time.

The convergence and accuracy of a method for solving high-order accurate bicompact schemes having the fourth order of approximation in spatial variables on a minimum stencil for a multidimensional inhomogeneous advection equation are investigated. The method is based on the approximate factorization of difference operators of multidimensional bicompact schemes. In addition, it uses iterations to preserve a high (higher than the second) order of accuracy of bicompact schemes in time. The convergence of these iterations for both two- and three-dimensional bicompact schemes as applied to the linear inhomogeneous advection equation with positive constant coefficients is proved using the spectral method. The efficiency of two parallel algorithms for solving equations of multidimensional bicompact schemes is compared. One of them is the spatial marching algorithm for calculating unfactorized schemes, and the other is based on iterative approximate factorization of difference operators of the schemes.

Рассматривается система уравнений Эйлера, описывающая многомерные течения невязкого многокомпонентного газа с несколькими химическими реакциями. Для этой системы методом расщепления по физическим процессам Марчука-Стрэнга строится неявная численная схема. Конвекция в ней рассчитывается по бикомпактной схеме SDIRK3B4 четвертого порядка по пространству и третьего порядка по времени, химические реакции - $L$-устойчивым методом Рунге-Кутты второго порядка. На одномерных и двумерных задачах с детонационными волнами проводится сравнение схемы SDIRK3B4 со схемой WENO5/SR. Показывается, что схема SDIRK3B4 при той же фактической точности обходится в 20-40 раз меньшим числом шагов по времени и не требует применения специальных процедур, подавляющих нефизический распад детонационных волн на относительно грубых сетках.

Работа посвящена использованию бикомпактных схем для численного решения эволюционных уравнений гиперболического типа. Основным преимуществом схем этого класса является сочетание двух положительных свойств: пространственной аппроксимации высокого четного порядка на шаблоне, всегда занимающем одну ячейку сетки, и спектрального разрешения, лучшего по сравнению с классическими компактными конечно-разностными схемами того же порядка пространственной аппроксимации. Рассматривается одна особенность бикомпактных схем — жесткая привязка их пространственной аппроксимации к декартовым сеткам (с ячейками-параллелепипедами в трехмерном случае). Она делает затруднительным применение бикомпактных схем к решению задач в сложных расчетных областях в рамках подхода неструктурированных сеток. Предлагается решать эту проблему путем применения известных методов аппроксимации границ сложной формы и соответствующих им краевых условий на декартовых сетках. Обобщение бикомпактных схем на задачи в геометрически сложных областях проводится на примере задач газовой динамики и уравнений Эйлера. В качестве конкретного метода, позволяющего учесть на декартовых сетках влияние твердых границ произвольной формы на течение газа, выбирается метод свободной границы. Приводится краткое описание этого метода, выписываются его уравнения. Для них строятся бикомпактные схемы четвертого порядка аппроксимации по пространству с локально-одномерным расщеплением. Компенсационный поток метода свободной границы дискретизируется со вторым порядком точности. Для интегрирования по времени в получаемых схемах применяются неявный метод Эйлера и L-устойчивый жестко-точный трехстадийный однократно диагонально-неявный метод Рунге–Кутты третьего порядка точности. Разработанные бикомпактные схемы тестируются на трех двумерных задачах: о стационарном сверхзвуковом обтекании с числом Маха, равным трем, одного круглого цилиндра и группы из трех круглых цилиндров, а также о нестационарном взаимодействии плоской ударной волны и круглого цилиндра в канале с плоскопараллельными стенками. Полученные результаты хорошо согласуются с результатами других работ: твердые тела физически корректно влияют на поток газа, давление в контрольных точках на поверхностях тел рассчитывается с точностью, в целом отвечающей выбранному разрешению сетки и уровню численной диссипации.

An implicit scheme with splitting with respect to physical processes is proposed for a stiff system of two-dimensional Euler gas dynamics equations with chemical source terms. For the first time, convection is computed using a bicompact scheme that is fourth-order accurate in space and third-order accurate in time. This high-order bicompact scheme is L-stable in time. It employs a conservative limiting method and Cartesian meshes with solution-based adaptive mesh refinement. The chemical reactions are computed using an L-stable second-order Runge–Kutta scheme. The developed scheme is successfully tested as applied to several problems concerning detonation wave propagation in a two-species ideal gas with a single combustion reaction. The advantages of bicompact schemes over the popular MUSCL and WENO5 schemes as applied to shock-capturing computations of detonation waves are discussed.

Bicompact schemes for multidimensional hyperbolic equations on Cartesian meshes with solution-based AMRHigh-order bicompact schemes for hyperbolic equations on Cartesian meshes with solution-based adaptive mesh refinement are constructed.The algorithm for implementation of these schemes on such meshes is described in detail.A new solution-based criteria of mesh refinement is proposed.Bicompact schemes with this refinement criteria are tested on the two-dimensional problem of compactly supported pulse advection and the two-dimensional Sedov blast wave problem.It is shown, that the design of bicompact schemes allows them to be implemented on meshes of such class with good accuracy of the computed solution ensured.

The Fourier analysis of fully discrete bicompact fourth-order spatial approximation schemes for hyperbolic equations is presented. This analysis is carried out on the example of a model linear advection equation. The results of Fourier analysis are presented as graphs of the dependence of the dispersion and dissipative characteristics of the bicompact schemes on the dimensionless wave number and the Courant number. The dispersion and dissipative properties of bicompact schemes are compared with those of other widely used difference schemes for hyperbolic equations. It is shown that bicompact schemes have one of the best spectral resolutions among the difference schemes being compared.

For the numerical solution of nonstationary quasilinear hyperbolic equations, a family of central semidiscrete bicompact schemes based on collocation polynomials is constructed in the one- and multidimensional cases. A dispersion analysis of semidiscrete bicompact schemes of fourth to eighth orders of accuracy in space is performed. Numerical examples are presented that demonstrate the ability of the bicompact schemes to adequately simulate wave propagation, including short waves, on highly nonuniform grids at long times. The properties of solutions of bicompact schemes in the problem of transfer of a stepwise initial profile are also considered.

A conservative limiting method for bicompact schemesIn this work, a new limiting method for bicompact schemes is proposed that preserves them conservative.The method is based upon a finite-element treatment of the bicompact approximation.An analogy between Galerkin schemes and bicompact schemes is established.The proposed method is tested on one-dimensional gasdynamics problems that include the Sedov problem, the Riemann "peak" problem, and the Shu-Osher problem.It is shown on these examples that bicompact schemes with conservative limiting are significantly more accurate than hybrid bicompact schemes.

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