Lie-group Shooting Method Research Articles

A Boundary-Type Numerical Procedure to Solve Nonlinear Nonhomogeneous Backward-in-Time Heat Conduction Equations

In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique.

A Non-Iteration Solution for Solving the Backward-in-Time Two-Dimensional Burgers’ equation with a Large Reynolds Number

This article proposes a noniteration solution based on the Lie-group shooting method (LGSM) to solve the backward-in-time two-dimensional Burgers’ equation with a large Reynolds number. The backward problem is famous for seriously ill-posed cases because the solution is generally unstable and highly dependent on the input data. Small perturbations in the input data, such as random errors inherent to the measurements in the analysis, can cause large oscillations in the solution. To handle a large Reynolds number under long time spans, it is very difficult to integrate towards the time direction. To avoid time integration and numerical iteration, the noniteration vector solution based on a two-point equation of the LGSM, including the initial and final conditions and boundary conditions (BCs) at the initial and terminal times, can be constructed. When the vector solution can be obtained from the ratios of the wave fronts on the BCs at the initial and terminal times, this solver can avoid the numerical iteration and numerical divergence of the conventional LGSM. Two benchmark examples in one and two variables are examined to illustrate the performance of the proposed method. The numerical results of this research are very consistent with the exact solutions when considering disturbances from noisy data. Even when the Reynolds number reaches 10E12, from the noisy final and boundary data, the noniteration solution can efficiently address the nonlinear Burgers’ problem with or without disturbances. This method does not use any transformation techniques, iterative processes, or regularization processes to avoid numerical instability. Hence, a noniterative solution is more stable and accurate for the unsteady nonlinear Burgers’ equation than currently used methods.

Simultaneous determination of the heat source and the initial data by using an explicit Lie-group shooting method

In this article, an explicit Lie-group shooting method (LGSM) is developed to solve the time-dependent heat source and the initial data for backward heat conduction problems. To recover both unknown data simultaneously, it is very difficult to obtain a stable solution by explicit or implicit schemes. To solve these problems by using conventional numerical schemes, numerical iterative regularization techniques and numerical integration techniques are necessary. To avoid these numerical techniques and to increase the computational efficiency, an explicit LGSM is developed. According to the solution of the quadratic equation of the LGSM, the initial condition can be directly obtained by using the final condition and boundary conditions at the initial time and final time. Using the reciprocal relationship of the solutions for the initial condition and the final condition, the proposed algorithm can avoid numerical integration and numerical iteration. Additionally, a closed-form formula from a two-point Lie-group equation can be directly used to calculate the heat source term. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks are tested. The numerical results indicate that the proposed algorithm can achieve an efficient and stable solution, even with noisy measurement data, by comparing the estimation results with the existing literature.

Estimation of space-dependent thermophysical properties in participating media by using a Lie-group shooting method

Estimation of space-dependent thermal conductivity and absorption coefficient in a one-dimensional participating medium was investigated in this study. For the forward problem, discrete ordinate method was used to solve the radiative transfer equation. The conduction-radiation energy equation was decoupled by the radiative source term and solved by the finite difference method. Lie-group shooting method (LGSM) was introduced to retrieve the space-dependent thermal conductivity and absorption coefficient from the coupled optical (radiative intensity profile) and thermal (temperature profile) information. LGSM was employed to solve the inverse problem of combined conduction-radiation heat transfer for the first time. This method does not require priori information about the functional form of thermal properties and initial guesses can be selected randomly. Numerical examples of the thermal conductivity and absorption coefficient under different distributions were examined to show that the new approach is highly accurate and efficient, even for identifying highly discontinuous and oscillatory parameters. The estimation was conducted in the presence of the random measurement noise. However, the estimated results still showed high accuracy, robustness and low susceptibility to the shape of property distributions.

A Geometric Approach for Solving Troesch’s Problem

In this paper, the Lie-group shooting method (LGSM) which is a numerical geometric integrator, is applied to the intrinsically unstable Troesch’s problem. The calculated results illuminate the efficiency and precision of LGSM for this problem.

A Lie-group shooting method for reconstructing a past time-dependent heat source

We consider an inverse problem for the reconstruction of a past unknown time-dependent heat source H( t), t < t f , in a heat conduction equation T t ( x, t) = T xx ( x, t) + H( t) with the aid of an extra measurement of temperature gradient on the left-boundary, where a final time condition is measured at the present terminal time t f . This inverse problem is quite difficult to be solved numerically owing to a twofold ill-posedness, as a combination of the backward heat conduction problem and the inverse heat source identification problem, which is abbreviated as inverse heat source/backward heat conduction problem (IHSBHCP). The new method proposed here, namely the Lie-group shooting method (LGSM), is examined through the tests by several numerical examples. Although the recovery of an unknown heat source is carried out under a presently measured temperature at a final time and under a large measurement noise both imposed on the final time data and all boundary data, the LGSM still works effectively and accurately. The accuracy in the reconstruction of H( t) is almost uneffected by different levels of noise.

The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems

A new computational method for solving the second-order nonlinear singularly perturbed boundary value problems (SPBVPs) is provided in this paper. In order to overcome a highly singular behavior very near to the boundary as being not easy to treat by numerical method, we adopt a coordinate transformation from an x-domain to a t-domain via a rescaling technique, which can reduce the singularity within the boundary layer. Then, we construct a Lie-group shooting method (LGSM) to search a missing initial condition through the finding of a suitable value of a parameter r ∈ [0, 1]. Moreover, we can derive a closed-form formula to express the initial condition in terms of r, which can be determined properly by an accurate matching to the right-boundary condition. Numerical examples are examined, showing that the present approach is highly efficient and accurate.

Reconstruction of moving boundary for one-dimensional heat conduction equation by using the Lie-group shooting method

We consider an inverse problem for the recovery of an unknown time-dependent moving boundary ξ(t) in a one-dimensional heat conduction equation. By using the domain embedding technique, the inverse problem of moving boundary identification is transformed into a time-dependent parameter function identification problem of an advection–diffusion partial differential equation, where ξ(t) and play the role of unknown parameters. From a viewpoint of numerical differentiation, that appearing in the governing equation causes the identification of ξ(t) being rather difficult, because the differential itself is an ill-posed operator. In order to overcome this difficulty we let , instead of ξ(t), be the unknown variable, and ξ(t) is calculated from by an integration. Once using the Lie-group shooting method (LGSM) we can derive a quite simple system of linear algebraic equations to iteratively calculate and then ξ(t). It is demonstrated through numerical examples that the LGSM is accurate, efficient and stable; the maximum error of numerical solutions is in the order of 10−4–10−2, even a large noise in the level of 10−2 is imposed on the measured data.

The Lie-group shooting method for solving the Bratu equation

For the Bratu problem, we transform it into a non-linear second order boundary value problem, and then solve it by the Lie-group shooting method (LGSM). LGSM allows us to search a missing initial slope and moreover, the initial slope can be expressed as a function of r ∈ [0, 1], where the best r is determined by matching the right-end boundary condition. The calculated results as compared with those calculated by other methods, illuminate the efficiency and precision of Lie-group shooting method (LGSM) for this problem.

Solving two typical inverse Stefan problems by using the Lie-group shooting method

We consider two typical inverse Stefan problems: one is computing a heat flux boundary condition when the moving boundary ξ( t) is given, and another is recovering an unknown moving boundary ξ( t), by knowing an extra Dirichlet boundary condition on the accessible boundary. Through a domain embedding method, we can transform the inverse problem into a parameter identification problem of an advection–diffusion partial differential equation, where ξ( t) and ξ ˙ ( t ) play the role of unknown parameters for the second inverse problem. The ξ ˙ ( t ) appeared in the governing equation makes the identification of ξ( t) rather difficult. However, upon using the Lie-group shooting method (LGSM) we can derive a simple system of algebraic equations to iteratively calculate ξ ˙ ( t ) and then ξ( t) at some discretized times. It is demonstrated through numerical examples that the LGSM is accurate and stable, although under a large measurement noise.

A highly accurate LGSM for severely ill-posed BHCP under a large noise on the final time data

We consider a backward heat conduction problem (BHCP) in a slab, by subjecting to data at a final time, and some different type boundary conditions at two ends of the slab. The BHCP is known to be highly ill-posed. In order to numerically solve the BHCP we develop a new Lie-group shooting method (LGSM) in the spatial direction. It can retrieve very well the initial data with a high order accuracy. Several numerical examples of the BHCP demonstrate that the LGSM is applicable, even for those of strongly ill-posed ones with a large value of final time. Under the noisy final data the LGSM is robust against the disturbance. The new method is applicable for a severely ill-posed case with a final data very small in the order of 10 −43, and the noise level is in the order of 10 −1, of which the numerical solution still has an accuracy in the order of 10 −2. The results are rather significant in the computations of BHCP.

A new shooting method for quasi‐boundary regularization of multi‐dimensional backward heat conduction problems

We employ a quasi‐boundary regularization to construct a two‐point boundary value problem for multi‐dimensional backward heat conduction equations. The multidimensional backward heat conduction problem (BHCP) is renowned as severely ill‐posed because the solution does not fullly depend on the data. In order to numerically tackle the multi‐dimensional BHCP, we propose a Lie‐group shooting method (LGSM) in the time direction to find the unknown initial conditions. The pivot point is based on the establishment of a one‐step Lie group element G(T) and the construction of a generalized mid‐point Lie group element G(r). Then, by imposing G(T) = G(r) we can search for the missing initial conditions through a minimum discrepancy to the real targets by the numerical ones, in terms of the weighting factor r ? (0, 1). When numerical examples are tested, we find that the LGSM is applicable to the BHCP. Even with noisy final data, the LGSM is also robust against disturbance.

A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature

We consider an inverse problem for estimating an unknown time-dependent heat source H ( t ) in a heat conduction equation T t ( x , t ) = T xx ( x , t ) + H ( t ) , with the aid of an extra measurement of temperature at an internal point. The Lie-group shooting method (LGSM) was used in the solution of this inverse problem; however, when the data are acquired at an internal point we require to develop a two-stage Lie-group shooting method (TSLGSM) to solve it. This novel approach is examined through numerical examples to convince that it is a rather accurate and efficient method, whose estimation error is small even for the identification of discontinuous and oscillatory heat source under large noise.

An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature

Considering here is an inverse problem for estimating the unknown nonhomogeneous heat conductivity function α ( x ) in T t ( x , t ) = ∂ ( α ( x ) T x ) / ∂ x + h ( x , t ) with the aid of an extra measurement of temperatures at a final time, which may be disturbed by random noise. A Lie-group shooting method (LGSM) is developed from the one-step Lie-group elements obtained by a spatial-discretization of heat conduction equation and by using the central difference or forward difference for α ′ ( x ) in spatial domain. The heat conductivities are available by directly solving linear equations. The new methods have twofold advantages in that no a priori information of heat conductivity is required and no iterations in the calculation process are needed. The accuracy and robustness of present methods are confirmed by comparing estimated results with exact solutions. The LGSM is stable and accurate, although the estimations are carried out under a large measurement noise.