Order Implicit Scheme Research Articles

Assessing the accuracy and efficiency of different order implicit and explicit integration schemes

A first order accurate fully implicit integration scheme and four different order explicit substepping integration schemes with automatic error control are used in this paper to integrate the constitutive relations of a critical state model for saturated soils. Their respective computational performance in terms of accuracy and efficiency is assessed in order to provide practical guidance for deciding which of the five is most suitable for solving numerical problems in geotechnical engineering involving critical state models. Even though existing literature of integration schemes applied to geotechnical problems has traditionally been focussed on the first order accurate implicit backward Euler and on the second order accurate explicit modified Euler with substepping almost exclusively, the findings of this paper suggest that the little extra work required in the implementation of an explicit third order Runge-Kutta substepping scheme is worth the effort, especially in terms of computational cost.

Temperature rise by viscous dissipation effect on synovial fluid induced by oscillating motion in artificial hip joint

A distinctive case study on the temperature rise, induced exclusively by viscous dissipation effect, due to the oscillating motion on a specific composition of synovial fluid within a clearance alike to an artificial hip joint, is investigated numerically for the first time, using the Ansys Fluent software. The joint clearance is modelled as concentric quarter hemispheres filled with a specific synovial fluid. The oscillating frequency considers the normal human walking speed and insulated thermal boundary conditions applied to surfaces that signifies the maximum viscous dissipation effect. Second order implicit time scheme is selected to achieve better result accuracy and ensure the unconditionally stable condition. The result shows that the viscous dissipation effect contributes to a maximum rise of 2.8° Celsius in temperature at the 1-h time mark, which is seen to lie within the in-vivo and in-vitro experiments reported in the literature. It is advocated that the viscous dissipation effect in the synovial fluid should be included in artificial hip joint model studies.

Quantifying hemodynamics within an aneurysm exposed to prolonged exercise levels

Background and objectiveNon-invasive treatment of unruptured Abdominal Aortic Aneurysm involves subjecting the patients to certain physiological levels of the heart. Flow topology (Repeak = 200–1200, frequency: f = 1.18–2.41 Hz) within an aneurysm geometry (2-D) under resting and exercise (mild and moderate) conditions are explored in the present study. Blood is assumed to be Newtonian in nature. Spatio-temporal evolution of the flow patterns and vorticity are established. Hemodynamic indicators (TAWSS and OSI), movement of vortex cores and Particle Residence Index (PRI) are quantified to select an optimum exercise level in attenuating the disease. MethodsThe finite volume method is employed for numerical solutions using ANSYS-FluentⓇ software. The SIMPLE scheme has been used for the pressure-velocity coupling. Least Square cell-based method is used for the spatial discretization of the gradients. Second order upwind scheme is considered for discretization of the pressure term. Third order upwind (QUICK) scheme is used to discretize the momentum equation. First order Implicit Scheme was used for the discretization of the temporal terms. Discrete Phase Material (DPM) technique is employed throughout, to visualize the signature of particle deposits within the aneurysm. ResultsVortex impingement induces a pressure peak within the aneurysm (moderate) while the peaks are anchored at the proximal and distal ends under resting and mild conditions. Along the averaged flow separation zone, exercise increases the maximum TAWSS from 1.21 N/m2 (mild) to 9.3 N/m2 (moderate). The distal site is exposed to oscillatory loading (OSI = 0.5) under mild activity whereas the loading becomes distributed almost over the entire wall, when subjected to moderate conditions. This in turn, reduces the time involved in 50 percent clearance of particles (PRI = 0.5) from 10.56 s (resting) to 3.98 s (mild) and 0.87 s (moderate), respectively. ConclusionsResting conditions manifests the aneurysmal wall to recirculating fluid for most of cycle time. Moderate exercise exhibits the least particle clearance time, but it exposes the aneurysmal wall and the distal end to high pressure, which otherwise has low intensity under mild activity. This in turn establishes that mild exercise for prolonged duration can be an optimum level for non-invasive aneurysmal treatment.

Verification of iterative matrix solutions for multipoint kinetics equations

Multipoint kinetic equation systems have been solved numerically using matrix algebra software and 1st to 4th order implicit schemes based on single-step matrix propagators. These matrix schemes have been validated successfully on demanding point kinetics benchmarks with various prescribed reactivity insertions. Verification tests have also been performed on a simple 3-region fast reactor core model with asymmetric step or ramp reactivity insertions and cross-checked with the multipoint kinetics SACRE code developed at INL and based on existing and validated stiff ODE solver packages. Accurate results can be obtained at a limited time expense. The reason for intriguing results obtained in some transients has been elucidated and linked to the multipoint matrix coefficient interpolation method during the transient.

Stability analysis of several first order schemes for the Oldroyd model with smooth and nonsmooth initial data

In this article, we develop several first order fully discrete Galerkin finite element schemes for the Oldroyd model and establish the corresponding stability results for these numerical schemes with smooth and nonsmooth initial data. The stable mixed finite element method is used to the spatial discretization, and the temporal treatments of the spatial discrete Oldroyd model include the first order implicit, semi‐implicit, implicit/explicit, and explicit schemes. The ‐stability results of the different numerical schemes are provided, where the first‐order implicit and semi‐implicit schemes are the ‐unconditional stable, the implicit/explicit scheme is the ‐almost unconditional stable, and the first order explicit scheme is the ‐conditional stable. Finally, some numerical investigations of the ‐stability results of the considered numerical schemes for the Oldroyd model are provided to verify the established theoretical findings.

Modeling of heavy oil-water core-annular upward flow in vertical pipes using the two-fluid model

In this work, a one-dimensional, isothermal, pseudo-transient two-fluid mathematical model is presented to simulate heavy oil-water core-annular upward flow. The model consists of mass, momentum and energy conservation equations for each phase, whose numerical solution is based on the finite difference technique with a first order implicit scheme. The model allows compute the profiles of pressure, velocity, volume fraction and temperature of the phases. Various water-wall and interfacial friction factors proposed in literature were evaluated. It was found that: (1) the interfacial friction factor has an important influence on pressure drop computation, (2) the interfacial shear stress is larger as the oil flow rate increases, and (3) to take into account the buoyancy effect, into the water-wall stress, improves the pressure computation. A characteristic analysis to determine the model stability was also developed and it was found that the well-posed region grows as the oil superficial velocity (Jo) increases, and the model is stable for Jo/Jw≥0.75 (Jw is the superficial water velocity). The model is in agreement with experimental data and other models reported in literature. The total pressure drop computations present deviations smaller than 6% from experimental data.

High order implicit and explicit Lie-group schemes for solving backward heat conduction problems

This paper proposes high order implicit and explicit Lie-group schemes for solving backward heat conduction problems (BHCPs). The solutions to BHCPs are generally unstable and highly dependent on input data. Small perturbations in the input data can cause large oscillations in the solution. To overcome this problem, the variables in the nonlinear and nonhomogeneous heat conduction equation are transformed to obtain a fictitious time coordinate system by introducing a fictitious time variable and viscosity-damping coefficient. The original government equation can be transformed into a new heat conduction equation of an evolution type. Because the conventional scheme combined with the group preserving scheme (GPS) in Minkowski space must satisfy the constraint of the cone structure, Lie group and Lie algebra at each fictitious time step, solutions highly dependent on ones cannot easily converge. We use an evolution-type equation to calculate in Euclidean space and develop high order implicit and explicit Lie-group schemes. More importantly, for this difficult BHCP, by using both schemes, the problem becomes a linear problem of a single parameter, without choosing the parameters, such as the viscosity-damping coefficient, fictitious time step and the fictitious terminal time. Additionally, we modify the high order explicit Lie-group scheme to reduce the number of iterations in the implicit Lie-group scheme. The accuracy and efficiency of both schemes are validated, even under noisy measurement data, by comparing the estimation results with previous literature.

Asymptotic analysis of discrete schemes for non-equilibrium radiation diffusion

Motivated by providing well-behaved fully discrete schemes in practice, this paper extends the asymptotic analysis on time integration methods for non-equilibrium radiation diffusion in 2 to space discretizations. Therein studies were carried out on a two-temperature model with Larsen's flux-limited diffusion operator, both the implicitly balanced (IB) and linearly implicit (LI) methods were shown asymptotic-preserving. In this paper, we focus on asymptotic analysis for space discrete schemes in dimensions one and two. First, in construction of the schemes, in contrast to traditional first-order approximations, asymmetric second-order accurate spatial approximations are devised for flux-limiters on boundary, and discrete schemes with second-order accuracy on global spatial domain are acquired consequently. Then by employing formal asymptotic analysis, the first-order asymptotic-preserving property for these schemes and furthermore for the fully discrete schemes is shown. Finally, with the help of manufactured solutions, numerical tests are performed, which demonstrate quantitatively the fully discrete schemes with IB time evolution indeed have the accuracy and asymptotic convergence as theory predicts, hence are well qualified for both non-equilibrium and equilibrium radiation diffusion. Provide AP fully discrete schemes for non-equilibrium radiation diffusion.Propose second order accurate schemes by asymmetric approach for boundary flux-limiter.Show first order AP property of spatially and fully discrete schemes with IB evolution.Devise subtle artificial solutions; verify accuracy and AP property quantitatively.Ideas can be generalized to 3-dimensional problems and higher order implicit schemes.

Computational gas dynamics in a wide range of Mach number on heterogeneous cluster architecture

The specific features of numerical method for gas dynamics problems are considered in this paper. The viscous flows are considered in a wide range of Mach number. The numerical method is based on conservation laws applied on unstructured heterogeneous mesh with further integration on complex cluster architectures. Application of unstructured mesh allows one to perform calculation of geometrically complex 3D objects with strict conservation properties. The developed numerical method includes multiple flux solvers (Godunov's, HLLC, Roe-Fix, AUSM, etc), first and second order spatial approximation, and explicit, semi-implicit or implicit first order time schemes. The method utilizes heterogeneous GPU HPC clusters. A wide range of problems is used for verification of the developed computational complex. All results on these problems had a good agreement with reference data.

A compact finite difference method for solving a class of time fractional convection-subdiffusion equations

A high-order compact finite difference method is proposed for solving a class of time fractional convection-subdiffusion equations. The convection coefficient in the equation may be spatially variable, and the time fractional derivative is in the Caputo’s sense with the order \(\alpha \) (\(0<\alpha <1\)). After a transformation of the original equation, the spatial derivative is discretized by a fourth-order compact finite difference method and the time fractional derivative is approximated by a \((2-\alpha )\)-order implicit scheme. The local truncation error and the solvability of the method are discussed in detail. A rigorous theoretical analysis of the stability and convergence is carried out using the discrete energy method, and the optimal error estimates in the discrete \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained. Applications using several model problems give numerical results that demonstrate the effectiveness and the accuracy of this new method.

A Posteriori Error Estimates for Leap-Frog and Cosine Methods for Second Order Evolution Problems

We consider second order explicit and implicit two-step time-discrete schemes for wave-type equations. We derive optimal order aposteriori estimates controlling the time discretization error. Our analysis, has been motivated by the need to provide aposteriori estimates for the popular leap-frog method (also known as Verlet's method in molecular dynamics literature); it is extended, however, to general cosine-type second order methods. The estimators are based on a novel reconstruction of the time-dependent component of the approximation. Numerical experiments confirm similarity of convergence rates of the proposed estimators and of the theoretical convergence rate of the true error.

Analysis and application of high order implicit Runge–Kutta schemes to collocated finite volume discretization of the incompressible Navier–Stokes equations

The application of a family of high order implicit Runge–Kutta time integration schemes, namely the explicit first-stage singly diagonally implicit Runge–Kutta schemes (ESDIRK), to cell-centered collocated finite volume discretization of the unsteady incompressible Navier–Stokes is considered. Although achieving computational efficiency relative to commonly used second order implicit schemes has been the motivating factor, this study focuses on temporal order analysis of the high order schemes on the collocated grid. In particular, a face velocity interpolation procedure (Rhie–Chow) which preserves the temporal design order of the ESDIRK schemes is introduced. The details of an iterative pressure-based time advancing algorithm comprising the designed interpolation method are presented (iterated-PISO). The results from solving numerical examples demonstrate the temporal order preservation of the algorithm.

An ALE Formulation for Explicit Runge–Kutta Residual Distribution

In this paper we consider the solution of hyperbolic conservation laws on moving meshes by means of an Arbitrary Lagrangian Eulerian (ALE) formulation of the Runge---Kutta Residual Distribution (RD) schemes of Ricchiuto and Abgrall (J Comput Phys 229(16):5653---5691, 2010). Up to the authors knowledge, the problem of recasting RD schemes into ALE framework has been solved with first order explicit schemes and with second order implicit schemes. Our resulting scheme is explicit and second order accurate when computing discontinuous solutions.

$$H^2$$ H 2 -Stability of the First Order Fully Discrete Schemes for the Time-Dependent Navier–Stokes Equations

This paper considers the $$H^2$$ H 2 -stability results for the first order fully discrete schemes based on the mixed finite element method for the time-dependent Navier---Stokes equations with the initial data $$u_0\in H^\alpha $$ u 0 ? H ? with $$\alpha =0,~1$$ ? = 0 , 1 and 2. A mixed finite element method is used to the spatial discretization of the Navier---Stokes equations, and the temporal treatments of the spatial discrete Navier---Stokes equations are the first order implicit, semi-implicit, implicit/explicit(the semi-implicit/explicit in the case of $${\alpha }=0$$ ? = 0 ) and explicit schemes. The $$H^2$$ H 2 -stability results of the schemes are provided, where the first order implicit and semi-implicit schemes are the $$H^2$$ H 2 -unconditional stable, the first order explicit scheme is the $$H^2$$ H 2 -conditional stable, and the implicit/explicit scheme (the semi-implicit/explicit scheme in the case of $${\alpha }=0$$ ? = 0 ) is the $$H^2$$ H 2 -almost unconditional stable. Moreover, this paper makes some numerical investigations of the $$H^2$$ H 2 -stability results for the first order fully discrete schemes for the time-dependent Navier---Stokes equations. Through a series of numerical experiments, it is verified that the numerical results are shown to support the developed $$H^2$$ H 2 -stability theory.

Analysis and application of high order implicit Runge–Kutta schemes for unsteady conjugate heat transfer: A strongly-coupled approach

Thermal interaction of fluids and solids, or conjugate heat transfer (CHT), is encountered in many engineering applications. Since time-accurate computations of unsteady CHT can be computationally demanding, we consider the use of high order implicit time integration schemes which have the potential to be more efficient relative to the commonly used second order implicit schemes. We present a strongly-coupled solution algorithm where the high order L-stable explicit first-stage singly diagonally implicit Runge–Kutta (ESDIRK) schemes are used to advance the solution in time within each separate fluid and solid subdomains. Furthermore, the stability and rate of convergence of performing (Gauss–Seidel) subiterations at each stage of the ESDIRK schemes are analyzed. The results from solving a numerical example (an unsteady conjugate natural convection in an enclosure) show good agreement with the performed analytical stability analysis. In addition, the (computational) work-(temporal) precision character of several schemes in solving a strongly coupled CHT problem is compared over a range of accuracy requirements. From the efficiency investigation, it is observed that performing subiterations with the strongly-coupled ESDIRK algorithm is more efficient than lowering time-step size using a high order loosely-coupled IMEX algorithm. In addition, by using the ESDIRK schemes, gain in computational efficiency relative to Crank–Nicolson is observed for time-accurate solutions (a factor of 1.4 using the fourth order ESDIRK). The computational gain is higher for smaller tolerances.

A high order time-accurate loosely-coupled solution algorithm for unsteady conjugate heat transfer problems

Thermal interaction of fluids and solids, or conjugate heat transfer (CHT), is encountered in many engineering applications. Noting that time-accurate computations of transient CHT problems can be computationally expensive, we consider the use of high order implicit time integration schemes which have the potential to be computationally more efficient relative to the commonly used second order implicit schemes. For thermally weak couplings, we present a loosely-coupled solution algorithm where high order implicit–explicit (IMEX) Runge–Kutta schemes are employed for time integration. The IMEX schemes consist of the explicit first-stage singly diagonally implicit Runge–Kutta (ESDIRK) schemes, for advancing the solution in time within each separate fluid and solid subdomain, and the explicit Runge–Kutta (ERK) schemes, for explicit integration of part of the coupling terms. By considering a numerical example (an unsteady conjugate natural convection in an enclosure), temporal order preservation of the coupling algorithm (without subiterating) is demonstrated. In addition, the stability of the loosely-coupled algorithm is investigated numerically for the CHT test-case; when the ratio of the thermal effusivities of the fluid and solid subdomains is much smaller than unity, using large Fourier numbers of the subdomains is possible, indicating that time-step size is restricted by accuracy rather than stability. Furthermore, the (computational) work-(temporal) precision character of several time integration schemes in solving the CHT test-case is compared over a range of accuracy requirements; for time-accurate solutions, the fourth and fifth order IMEX schemes are 1.5 times more efficient than Crank–Nicolson and 2.7 times more efficient than BDF2. The computational gain is higher for smaller tolerances.

Numerical investigation of Maxwell–Vlasov equations – Part I: Basic physics and algorithms

We present a numerical scheme to solve the fully 3D Maxwell–Vlasov equations. The aim is to investigate the interaction of a laser pulse with a plasma and the acceleration of electrons and ions. We choose the approach based on numerical particles to approximate the distribution function interpreted, according to the Klimontovich formalism, as a probability distribution function. The key points are a high order implicit scheme to approximate the space differential operators and a time evolution scheme of adequate accuracy.

Implicit Scheme for Hyperbolic Conservation Laws Using Nonoscillatory Reconstruction in Space and Time

The efficiency of high order accurate schemes for the solution of unsteady hyperbolic conservation laws is adversely affected by time-step restrictions that arise from monotonicity requirements. When applied to the solution of problems involving discontinuities, these restrictions render conventional high order implicit time integration schemes impractical. In the present study, a new single step second order implicit scheme that uses nonoscillatory reconstruction in space and time is presented. Both the spatial and temporal limiters are dependent on the evolving solution, and this nonlinearity allows for a circumvention of total variation diminishing bounds. Numerical results on model scalar and vector hyperbolic equations suggest that the scheme holds promise in achieving accurate and unconditionally nonoscillatory solutions.

532 NS 方程式によるバクテリアの推進機構の解析

The speed of a bacteria, which uses a rotating helical flagellum to produce propulsion, is calculated by solving the full Navier-Stokes equations. This rotating flagellum is treated by sliding mesh technique. The control volume method is utilized for discretizing the equations, and the first order scheme for the convection term, the second order scheme for the diffusion term, and the first order implicit scheme for the unsteady term. The speed is estimated by converting the kinetic energy produced by the rotating flagellum. It is found that higher rotational speed gives more conversion loss to the speed of bacteria, and that the order of the speed estimated by the present method agrees with that by the boundary element method with the Stokes approximation.

Three-dimensional incompressible viscoelasticity in large strains: Formulation and numerical approximation

This paper considers viscoelastic incompressible solids in large deformations. A three-dimensional thermodynamical model is first proposed. It leads to a nonlinear variational problem which is approximated in space by mixed finite elements and in time by a first order implicit scheme. This approximation is analyzed on the corresponding linearized problem. Finally, an elimination strategy is described which reduces each step of the discrete problem to the solution of a standard hyperelastic problem to be solved by Newton's method.