runge kutta solutions of a hyperbolic conservation law with source term spurious long term solutions of a finite difference method for a hyperbolic conservation law with a general nonlinear source term are studied results are contrasted with those that have been established for nonlinear ordinary differential equations various types of spurious behavior are examined including spatially uniform equilibria that exist for arbitrarily small time steps nonsmooth steady states with profiles that jump between fixed levels and solutions with oscillations that arise from nonnormality and exist only in finite precision arithmetic it appears that spurious behavior is associated in general with insufficient spatial resolution the potential for curbing spuriosity by using adaptivity in space or time is also considered

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

the 3 d elastodynamic boundary element method semi analytical integration for linear isoparametric triangular elements a semi analytical integration scheme is described in this paper which is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically this new scheme can be applied to linear triangular elements for use in steady state elastodynamic bem problems and is particularly useful for predicting displacement to high accuracy close to surfaces for a spectrum of frequencies the scheme involves the application of taylor expansions to formulate the integrals into two parts one part is regular and is evaluated numerically and the other part is singular but sufficiently simple to enable its transforma tion into a line integral the line integral is solved numerically using gauss legendre quadrature this approach caters for all the integral types that appear in steady state elastodynamic boundary elements but in particular no special treatment is required for the evaluation of the cauchy principal value singular integrals numerical tests are performed on a simple test problem for which a known analytical solution exists the results obtained using the semi analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods copyright 1999 john wiley sons ltd

a new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method the purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto plastic solids respectively by the direct boundary element method bem triangle polar co ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares the method was subsequently extended to the cubature of singular integrals over three dimensional internal cells as occur in applications of the bem to three dimensional elasto plasticity for this purpose so called tetrahedron polar co ordinates were introduced singular boundary integrals stretching over either linear triangular or quadratic quadilateral isoparametric boundry elements and singular volume integrals extending over either linear tetrahedral or quadratic hexahedral isoparametric internal cells are treated in case of higher order isoparametric boundary elements and internal cells division into a number of subelements and subcells respectively is necessary the analytical investigation is followed by a numerical study restricted to the use of quadratic quadrilateral isoparametric boundary elements this is justified by the fact that such elements as opposed to linear elements yield singular boundary integrals which cannot be integrated analytically the results of the numerical investigation demonstrate the potential of the developed concept

generic domain decomposition and iterative solvers for 3d bem problems summary in the past two decades considerable improvements concerning integration algorithms and solvers involved in boundary element formulations have been obtained first a great deal of efficient techniques for evaluating singular and quasi singular boundary element integrals have been definitely established and second iterative krylov solvers have proven to be advantageous when compared to direct ones also including non hermitian matrices the former fact has implied in cpu time reduction during the assembling of the system of equations and the latter fact in its faster solution in this paper a triangle polar co ordinate transformation and the telles co ordinate transformation applied in previous works independently for evaluating singular and quasi singular integrals are combined to increase the efficiency of the integration algorithms and so to improve the performance of the matrixassembly routines in addition the jacobi preconditioned biconjugate gradient j bicg solver is used to develop a generic substructuring boundary element algorithm in this way it is not only the system solution accelerated but also the computer memory optimized discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving in the discussion of results expressed in terms of accuracy and cpu time advantages and potential applications of the be code developed are highlighted copyright 2006 john wiley sons ltd

upper semicontinuity of attractors for approximations of semigroups and partial differential equations abstract suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined applications are given to parabolic and hyperbolic partial differential equations

efficient local energy dissipation preserving algorithms for the cahn hilliard equation abstract in this paper we show that the cahn hilliard equation possesses a local energy dissipation law which is independent of boundary conditions and produces much more information of the original problem to inherit the intrinsic property we derive three novel local structure preserving algorithms for the 2d cahn hilliard equation by the concatenating method in particular when the nonlinear bulk potential f in the equation is chosen as the ginzburg landau double well potential the method discussed by zhang and qiao 2012 50 is a special case of our scheme ii thanks to the leibnitz rules and properties of operators the three schemes are rigorously proven to conserve the discrete local energy dissipation law in any local time space region under periodic boundary conditions the schemes are proven to possess the discrete mass conservation and total energy dissipation laws numerical experiments are conducted to show the performance of the proposed schemes

constructing exact significance tests with restricted randomization rules summary in a clinical trial comparing two treatments suppose that subjects are assigned sequentially to the treatment groups by a restricted randomization rule under the randomization model an efficient recursive algorithm is provided to generate the exact permutational distribution for linear rank statistics given the final imbalance between the number of patients in the two groups if only the significance level of the test is required the efficiency of the algorithm can be improved significantly in general by the time the size of the trial reaches 30 the large sample approximation to the permutation distribution wei smythe smith 1986 is satisfactory the new procedure is illustrated by censored survival data from a prostatic cancer clinical trial we show that if the study subjects are not a random sample from a well defined population then the treatment allocation rule used at the design stage should not be ignored in the analysis

randomization matching and propensity scores in the design and analysis of experimental studies with measured baseline covariates in many experimental situations researchers have information on a number of covariates prior to randomization this information can be used to balance treatment assignment with respect to these covariates as well as in the analysis of the outcome data in this paper we investigate the use of propensity scores in both of these roles we also introduce a randomization procedure in which the balance of all measured covariates is approximately indexed by the variance of the empirical propensity scores and randomization is restricted to those permutations with the least variable propensity scores this procedure is compared with recently proposed methods in terms of resulting covariate balance and estimation efficiency properties of the estimators resulting from each procedure are compared with estimates which incorporate the propensity score in the analysis stage simulation results show that analytical adjustment for the propensity score yields results on par with those obtained through restricted randomization procedures and can be used in conjunction with such procedures to further improve inferential efficiency

grading functions and mesh redistribution we consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two point boundary value problems by finite difference or finite element methods an analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary value problems in one dimension error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria

adaptive pseudospectral solution of a diffuse interface model a diffuse interface type model using an energy based variational formulation with a free energy that is a function of the density and its gradients is presented all of the boundary terms are retained and related to external surface forces which can be of particular interest when considering the fluid fluid solid region the numerical solution of these types of problems can be troublesome if a thin transition layer is desired here chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied a mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived in order to achieve high precision for problems exhibiting thin transition layers a modified version of the arc length monitor function is proposed which yields a sufficiently smooth coordinate transformation at every step of the continuation process a fixed number of iterations is implemented so that the equidistribution equations are not solved completely at each step which saves a considerable amount of computational effort numerical results for the static phase field model exhibiting thin transition layers are presented

some improvements of accuracy and efficiency in three dimensional direct boundary element method numerical analysis with the boundary element method bem has been used more and more in various engineering fields in recent years in numerical techniques however there are some problems which have not been fully solved even now the most essential one is the drop in the accuracy of results for internal points near the boundary of the structure where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method for the boundary integral equation of stress this problem became more serious and the accuracy can be improved only partly even though very refined boundary elements are used in this paper the boundary integral equation is newly formulated using a relative quantity of displacement in this way the singularity of boundary integrals is reduced by the order of 1 r and the accuracy of solution is improved significantly furthermore in order to integrate it more accurately two kinds of numerical integral methods are newly developed by using these methods both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty although the discretization may be comparatively coarse the generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems

a general and effective way for evaluating the integrals with various orders of singularity in the direct boundary element method a general and efficient technique is developed for the evaluation of the integrals with various orders of singularity such as occur in the three dimensional boundary element method bem generalized extended triangle tetrahedron polar co ordinate mappings together with two conditions are used to remove the singularity of the integrals and to evaluate the corresponding non singular ones in a new numerical space triangle and tetrahedron polar co ordinates in reference 1 are proved to be a special case of the generalized ones in this paper with the developed idea boundary element results converge rapidly towards the analytical solutions for the strongly singular integrals evaluated directly and the analytical solutions can be gained in principle even when employing higher order triangular boundary elements and tetrahedral cells the generality and practicability of the method are demonstrated in the case of higher order elements discontinuous elements and large engineering problems

computer evaluation of singular integrals and their applications an algorithm and a standard subroutine are developed for the evaluation of singular integrals over arbitrary two dimensional domains the integrand is a product of a green s function with another function having a weak singularity at the boundary of the domain formulae are derived for an accurate estimation of the integral in the neighbourhood of the singularities the integral over the rest of the domain is evaluated by a library subroutine the software developed is applied to a study of some contact problems of the theory of elasticity for non classical domains

constrained randomization in a therapeutic efficacy trial the use of valid constrained randomization is presented for a therapeutic efficacy trial the construction and evaluation of randomization schemes are studied for two treatment replicates per block and for hypothesized linear gradients two different isomorphism classes of designs are compared using the variance of the mean squared error criterion the results indicate that valid constrained randomization schemes can be constructed that are superior to ordinary randomization a mastitis efficacy trial provided the specific problem to be solved

second order schemes and time step adaptivity for allen cahn and cahn hilliard models abstract in this paper we focus on efficient second order in time approximations of the allen cahn and cahn hilliard equations first of all we present the equations generic second order schemes based on a mid point approximation of the diffusion term and some schemes already introduced in the literature then we propose new ways of deriving second order in time approximations of the potential term starting from the main schemes introduced in guillen gonzalez and tierra 2013 yielding to new second order schemes for these schemes and other second order schemes previously introduced in the literature we study the constraints on the physical and discrete parameters that can appear to assure the energy stability unique solvability and in the case of nonlinear schemes the convergence of newton s method to the nonlinear schemes moreover in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step finally we compare the behaviour of the schemes and the effectiveness of the adaptive time stepping algorithm through several computational experiments

a method for direct evaluation of singular integral in direct boundary element method a new method is presented for evaluating the cauchy principal value integral and the free term in the three dimensional elastic boundary element method this method expresses the singular integral as the sum of a weakly singular integral and an integrable cauchy principal value integral the former can be computed by means of the usual standard gaussian quadrature after a transformation suggested by lachat and watson 10 who correctly limited its use only to weakly singular integral the latter can be calculated by direct integration in the cauchy principal value sense this paper also points out some limitations in the use of triangle polar co ordinate suggested by li et al 11 and later extended by zhang and xu 16 the general expression of the free term for a boundary point with arbitrary characteristic surface is also presented simple numerical examples which verify the efficiency and accuracy of the present method are given

the numerical computation of heteroclinic connections in systems of gradient partial differential equations the numerical computation of heteroclinic connections in partial differential equations pdes with a gradient structure such as those arising in the modeling of phase transitions is considered initially a scalar reaction diffusion equation is studied structural assumptions are made on the problem to ensure the existence of absorbing sets and consequently a global attractor as a result of the gradient structure it is known that if all equilibria are hyperbolic the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits results are proved about the fourier spanning basis for branches of equilibria and also for certain heteroclinic connections these results exploit the oddness of the nonlinearity the reaction diffusion equation is then approximated by a galerkin spectral discretization to produce a system of ordinary differential equations

finite difference approximations of multidimensional convection diffusion reaction problems with small diffusion on a special grid a numerical scheme for the convection diffusion reaction cdr problems is studied herein we propose a finite difference method on a special grid for solving cdr problems particularly designed to treat the most interesting case of small diffusion we use the subgrid nodes in the link cutting bubble lcb strategy 5 to construct a numerical algorithm that can easily be extended to the higher dimensions the method adapts very well to all regimes with continuous transitions from one regime to another we also compare the performance of the present method with the streamline upwind petrov galerkin supg and the residual free bubbles rfb methods on several benchmark problems the numerical experiments confirm the good performance of the proposed method

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

the global dynamics of discrete semilinear parabolic equations a class of scalar semilinear parabolic equations possessing absorbing sets a lyapunov functional and a global attractor are considered the gradient structure of the problem implies that provided all steady states are isolated solutions approach a steady state as t to infty the dynamical properties of various finite difference and finite element schemes for the equations are analysed the existence of absorbing sets bounded independently of the mesh size is proved for the numerical methods discrete lyapunov functions are constructed to show that under appropriate conditions on the mesh parameters numerical orbits approach steady state solutions as discrete time increases however it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented furthermore it is also shown that the explicit euler scheme introduces spurious solutions with period 2 in the timestep as a result the absorbing set is destroyed and there is initial data leading to blow up of the scheme however small the mesh parameters are taken to obtain stabilization to a steady state for this scheme it is necessary to restrict the timestep in terms of the initial data and the space step implicit schemes are constructed for which absorbing sets and lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step both one step and multistep bdf methods are studied

some notes on singular integral techniques in boundary element analysis the paper presented here reviews the numerical techniques used currently to calculate the singular integrals and nearly singular integrals in the boundary element analysis some incorrect algorithms published before are discussed and a new numerical technique to calculate the nearly singular integral is developed the numerical results show a significant improvement in both accuracy and efficiency compared to the traditional adaptive gaussian quadrature and subdivision techniques

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

error analysis of full discrete invariant energy quadratization schemes for the cahn hilliard type equation abstract in this paper we present the error analysis for a fully discrete scheme of the cahn hilliard type equation along with numerical verifications the numerical schedule is developed by first transforming the cahn hilliard type equation into an equivalent form using the invariant energy quadratization ieq technique then the equivalent form is discretized by using the linear implicit crank nicolson method for the time variable and the fourier pseudo spectral method for the spatial variables the resulted full discrete scheme is linear and unconditionally energy stable which makes it easy to implement by constructing an appropriate interpolation equation the uniform boundedness of the numerical solution is obtained theoretically then we prove that the numerical solutions converge with the order o t 2 h m where t is the temporal step and h is the spatial step with m the regularity of the exact solution several numerical examples are presented to confirm the theoretical results and demonstrate the effectiveness of the presented linear scheme the numerical strategies and analytical tools in this paper could be readily applied to study other phase field models or gradient flow problems

estimating quadrature errors for an efficient method for quasi singular boundary integral abstract the numerical resolution of the boundary integral equations applied to the differential equations of laplace helmholtz and maxwell requires the handling of quasi singular integrals with different order of singularity the numerical approximation of the integral equations of different kinds is made by boundary finite elements in this paper we present a complete survey for estimating quadrature errors for the numerical techniques proposed by huang and cruse q huang t a cruse some notes on singular integral techniques in boundary element analysis int j numer methods eng 36 15 1993 2643 2659 to calculate the quasi singular integrals to validate the accuracy and efficiency of these techniques and approve our study some numerical examples are presented and discussed

steady shock tracking and newton s method applied to one dimensional duct flow abstract a new computational approach combining shock tracking and newton s method is applied to steady one dimensional flow through a variable area duct on a transformed computational grid where the shock is fixed the physical shock location appears explicitly as an unknown in a set of finite difference equations and is coupled to the other unknowns the space time characteristics for the associated time dependent problem are used in formulating the boundary conditions the resulting system is solved by newton s method the computed results agree very well with an exact solution and the newton iterates converge rapidly in comparison to some explicit shock capturing time asymptotic methods

internal layers small eigenvalues and the sensitivity of metastable motion on a semi infinite domain an analytical characterization of exponentially slow internal layer motion for the allen cahn equation and for a singularly perturbed viscous shock problem is given the results extend some previous results that were restricted to a finite geometry for these slow motion problems we show that the slow dynamics associated with the semi infinite domain are not preserved even qualitatively by imposing a commonly used form of artificial boundary condition to truncate the semi infinite domain to a finite domain this extreme sensitivity to boundary conditions and domain truncation is a direct result of the exponential ill conditioning of the underlying linearized problem for burgers equation many of the analytical results are verified by calculating certain explicit solutions some related ill conditioned internal layer problems are examined

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

on the solution of convection diffusion boundary value problems using equidistributed grids the effect of using grid adaptation on the numerical solution of model convection diffusion equations with a conservation form is studied the grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc length of the solution in particular a parameter dependent monitor function is introduced which incorporates fixed meshes approximate arc length equidistribution and equidistribution of the absolute value of the solution in a single framework thus the resulting numerical method is a coupled nonlinear system of equations for the mesh spacings and the nodal values a class of singularly perturbed problems including burgers s equation in the limit of small viscosity is studied singular perturbation and bifurcation techniques are used to analyze the solution of the discretized equations and numerical results are compared with the results from the analysis computation of the bifurcation diagram of the system is performed numerically using a continuation method and the results are used to illustrate the theory it is shown that equidistribution does not remove spurious solutions present on a fixed mesh and that furthermore the spurious solutions can be stable for an appropriate moving mesh method

modified arc length adaptive algorithms for degenerate reaction diffusion equations this paper is concerned with modified adaptive difference schemes for solving degenerate nonlinear reaction diffusion equations with singular source terms the adaptive mechanisms both in space and time are considered modified monitor functions based on arc lengths of the rate function u t are constructed further analyses are given on properties of the numerical schemes derived we show that under proper smoothness consistency and step size constraints the numerical solution obtained preserves the monotonicity of the physical solution numerical experiments are given to demonstrate the monotonicity and convergence properties of the methods

restricted randomization a practical example abstract a method of restricted randomization which avoids bad patterns in the treatments but retains validity in the analysis is demonstrated on an example that exhibits many features of real agricultural experiments such as untidy treatment structure and uncertainty about blocking the method uses permutation groups modern farming methods force agricultural experimenters to use long thin lines of plots for blocking rather than more compact areas ordinary randomization of a long thin block can produce a plan with say all low levels of nitrogen at one end of the block the first section of the article discusses why simple rerandomization is not a satisfactory method of avoiding poor layouts restricted randomization has been developed as a method of avoiding badly patterned plans while ensuring validity of the usual analysis one method of restricted randomization uses special groups of permutations of the plots catalogs of good restricted randomizations have been given for symmetric factorial de

shock layer movement for burgers equation this paper considers the motion of the shock layer exhibited by the solution of the initial boundary value problem for the equation epsilon u xx uu x u t with small values of epsilon 0 this burgers equation occurs as a model for a number of physical problems and is representative of many convection dominated evolution equations a specific example will be solved explicitly using the cole hopf transformation the result suggests a corresponding asymptotic approach which provides the shock layer location x epsilon t as the solution of an initial value problem that method which can be applied to a far wider class of equations also shows how the steady state shock location can be changed substantially by asymptotic exponentially small perturbations

a novel linear unconditional energy stable scheme for the incompressible cahn hilliard navier stokes phase field model abstract in this paper we propose a linear unconditional energy stable time discretization scheme for cahn hilliard navier stokes model which is a phase field model for two phase incompressible flow based on a lagrange multiplier approach our proposed scheme is linearized by using implicit explicit treatments and the computation of the velocity field u the pressure p the phase function are decoupled using the mathematical induction the bound n l is obtained rigorous error analysis for the variable u p and are carried out in semi discrete form finally some numerical experiments are done to demonstrate the effectiveness for the proposed method