The successful use of mono-implicit Runge-Kutta methods has been demonstrated by several research... more The successful use of mono-implicit Runge-Kutta methods has been demonstrated by several researchers who have employed these methods in software packages for the numerical solution of boundary value ordinary di erential equations. However, these methods are only applicable to rst order systems of equations while many boundary value systems involve higher order equations. While it is straightforward to convert such systems to rst order, several advantages, including substantial gains in e ciency, higher continuity of the approximate solution and lower storage requirements, are realized when the equations can be treated in their original higher order form. In this paper, we consider generalizations of mono-implicit Runge-Kutta methods, called monoimplicit Runge-Kutta-Nystrom methods, suitable for systems of second order ordinary di erential equations, having the general form, y 00 (t) = f(t; y(t); y 0 (t)), and derive optimal symmetric methods of orders two, four, and six. We also in...
The BACOL software package, which employs high order methods in time and space to adaptively cont... more The BACOL software package, which employs high order methods in time and space to adaptively control spatial and temporal errors in a method-of-lines approach, has been shown to be significantly more efficient than existing codes for the accurate numerical solution of systems of parabolic PDEs in one space dimension. In BACOL, the collocation spatial discretization gives a system of differential-algebraic equations (DAEs) which is treated using the DAE solver DASSL. Since DASSL employs backward differentiation formulas (BDFs), each spatial remeshing requires an interpolation of previous solution values. In addition, for DAE systems whose Jacobians have eigenvalues on the imaginary axis, such as those arising from Schrödinger problems, DASSL performs inefficiently since the higher order BDFs are not A-stable. In this paper, we describe a new software package, BACOLR, which addresses these issues by using RADAU5, a DAE solver based on an A-stable, (one-step) implicit RungeKutta method...
BACOL is a software package for the numerical solution of systems of one-dimensional parabolic pa... more BACOL is a software package for the numerical solution of systems of one-dimensional parabolic partial differential equations (PDEs) that has been shown to be superior to other similar packages, especially for problems exhibiting sharp spatial layer regions where a stringent tolerance is imposed. BACOL, based on a method-of-lines algorithm, features adaptive control of a high order estimate of the spatial error. (Adaptive control of the temporal error in the numerical solution of the system of differential-algebraic equations (DAEs), arising from a B-spline Gaussian collocation spatial discretization, is provided by the underlying DAE solver, DASSL.) The spatial error estimate for the collocation solution computed by the code is obtained by computing a second collocation solution, which involves a substantial cost the execution time and memory usage are almost doubled. In this report we discuss BACOLI, a new version of BACOL that computes only one collocation solution and uses effic...
BACOL and BACOLR are B-spline Gaussian collocation method-of-lines packages for the numerical sol... more BACOL and BACOLR are B-spline Gaussian collocation method-of-lines packages for the numerical solution of systems of one-dimensional parabolic partial differential equations (PDEs). In previous studies, they were shown to be superior to other similar packages, especially for problems exhibiting sharp spatial layer regions where a stringent tolerance is imposed. A significant feature of these solvers is that, in addition to the temporal error control provided by the underlying time-integrator, they adapt the spatial mesh to control a high order estimate of the spatial error. In addition to computing a primary collocation solution of a given spatial order, the BACOL/BACOLR codes also compute, at a substantial cost, a secondary collocation solution of one higher order, and then the difference between the two collocation solutions is used to give an estimate of the leading order term in the error for the lower order solution. In this paper we consider an approach in which the computatio...
Error control software packages based on Gaussian collocation have been widely used for the numer... more Error control software packages based on Gaussian collocation have been widely used for the numerical solution of boundary value ODEs (BVODEs) and 1D parabolic time-dependent PDEs (1D PDEs) for several decades. These robust and efficient packages are among the best available for these problem classes. In this paper, we survey error control Gaussian collocation software for BVODEs and 1D PDEs and provide an overview of recent work involving the development of two new packages, one for each problem class. The first is an updated version of the well-known COLSYS/COLNEW package for BVODEs. The second is the newest member of the BACOL family of software packages for 1D PDEs. We briefly review the underlying numerical algorithms employed in these packages and then provide numerical results that show the superiority of the new packages compared to previously released packages from each software class.
Computers & Mathematics with Applications, 2019
B-spline Gaussian collocation software has been widely used in the numerical solution of boundary... more B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and partial differential equations (PDEs) in one space dimension (1D) for many years. The software package, BACOL, developed over a decade ago, was one of the first 1D PDE packages to provide both temporal and spatial error control. A new package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages implies that extensive testing and analysis of the test results is an essential factor in their development. In this paper, we investigate the performance of the BACOL and BACOLI packages with respect to several important machine independent algorithmic measures and examine the effectiveness of the new error estimation and error control strategies. We also investigate the influence of the choice of the degree of the B-splines on the efficiency and reliability of the solvers. These results will provide new insights into how to improve BACOLI, lead to improvements in the Gaussian collocation BVODE solvers, COLSYS and COLNEW, and guide the further development of B-spline Gaussian collocation software with error control for 2D PDEs.
This paper considers the use of a superconvergent interpolant (SCI) for spatial error estimation ... more This paper considers the use of a superconvergent interpolant (SCI) for spatial error estimation when Gaussian collocation is employed as the spatial discretization scheme in a method-of-lines algorithm for the numerical solution of a system of one-dimensional parabolic partial differential equations (PDEs). Gaussian collocation is a popular approach for the spatial discretization of parabolic PDEs, and at certain points within the problem domain, the collocation solution is superconvergent. This paper describes how an interpolant based on these superconvergent values can be used to provide an efficient error estimate for the collocation solution. We implement this scheme within a modified version of the collocation PDE solver, BACOL. The original BACOL code obtains a spatial error estimate by computing a second global collocation solution of one higher order of accuracy. We show that the SCI based error estimation approach can provide spatial error estimates of comparable accuracy to those currently computed by BACOL, but at a much lower cost.
We consider systems of parabolic partial differential equations in one space variable, for which ... more We consider systems of parabolic partial differential equations in one space variable, for which we describe a method of lines algorithm based on monomial spline collocation for the discretization of the spatial domain. While the usual application of this technique transforms the system of partial differential equations into a system of time dependent ordinary differential equations which can be integrated by standard initial value solvers, our approach leads to coupled systems of differential-algebraic equations. These equations are solved using a well known package, DASSL, which we have modified to take advantage of the special structure of the Jacobians which arise.
Advances in Applied Mathematics and Mechanics, 2013
In this paper we describe newB-spline Gaussian collocation software for solving two-dimensional p... more In this paper we describe newB-spline Gaussian collocation software for solving two-dimensional parabolic partial differential equations (PDEs) defined over a rectangular region. The numerical solution is represented as a bi-variate piecewise polynomial (using a tensor productB-spline basis) with time-dependent unknown coefficients. These coefficients are determined by imposing collocation conditions: the numerical solution is required to satisfy the PDE and boundary conditions at images of the Gauss points mapped onto certain subregions of the spatial domain. This leads to a large system of time-dependent differential algebraic equations (DAEs) which is solved using the DAE solver, DASPK. We provide numerical results in which we use the new software, called BACOL2D, to solve three test problems.
A long-standing open question associated with the use of collocation methods for boundary value o... more A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obtained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h 2k), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C 0 continuous solution approximation that has a global error of O(h k+1). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C 1 continuous "superconvergent" interpolants whose global errors are O(h 2k). The key ideas are to use the theoretical framework of continuous Runge-Kutta schemes and to augment the collocation solution with inexpensive monoimplicit Runge-Kutta stages. Specific schemes are derived for k = 1, 2, 3, and 4. Numerical results are provided to support the theoretical analysis.
Proceedings of the 21st Koli Calling International Conference on Computing Education Research, 2021
Here we describe a project whose goal is to address issues associated with student takeaway, i.e.... more Here we describe a project whose goal is to address issues associated with student takeaway, i.e., enduring learning, in an introductory course in Numerical Analysis/Scientific Computing (NA/SC) commonly taught in undergraduate programs in Computer Science, Mathematics, and Engineering. We have employed the well-known framework of Threshold Concepts (TCs) in order to identify essential "takeaway" concepts in an introductory NA/SC course. We report on the four TCs we have proposed for NA/SC and discuss how the TC framework can be used to improve student takeaway.
Recent investigations of discretization schemes for the efficient numerical solution of boundary ... more Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C 1 continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. N...
. The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (I... more . The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODE's more than fifteen years ago. During the last decade, there has been considerable investigation of the use of these schemes in the numerical solution of boundary value ODE problems, where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes. In fact, recent work in this area has seen the development of some software packages for boundary value ODE's based on these schemes. Unfortunately, these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval, while the collocation schemes provide a natural continuous solution approximation. The availability of a continuous solution is important not only to the user of the software but also within the code itse...
A long-standing open question associated with the use of collocation methods for boundary value o... more A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, ob-tained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h 2k), where k is the number of collocation points used per subinterval and h is the subin-terval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C 0 continuous solution approximation that has a global error of O(h k+1). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C 1 continuous "superconvergent" interpolants whose global errors are O(h 2k). The key ideas are to use the theo-retical f...
A popular approach to the numerical solution of boundary value ODE problems involves the use of c... more A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge-Kutta formulas, known as mono-implicit Runge-Kutta (MIRK) formulas, which can be implemented at a lower cost per step than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather, only a discrete approximation at certain points within the problem interval is obtained. However, recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge-Kutta formula. These ideas have recently been extended to develop continuous extensions of the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on the continuous MIRK formulas, as an alternative to the standard use of global error control, as the basis for termination and mesh redistribution criteria.
The defect of a continuous approximate solution to an ODE is the amount by which that approximati... more The defect of a continuous approximate solution to an ODE is the amount by which that approximation fails to satisfy the ODE. A number of studies have explored the use of asymptotically correct defect estimates in the numerical solution of initial value ODEs (IVODEs). By employing an appropriately constructed interpolant to an approximate discrete solution to the ODE, various researchers have shown that it is possible to obtain estimates of the local error and/or the maximum defect that are asymptotically correct on each step, as the stepsize h → 0. In this paper, we investigate the usefulness of asymptotically correct defect estimates for defect control in boundary value ODE (BVODE) codes. In the BVODE context, for a sequence of meshes which partition the problem interval, one computes a discrete numerical solution, constructs an interplant, and estimates the maximum defect. The estimates (typically obtained by sampling the defect at a small number of points on each subinterval of the mesh) are used in a redistribution process to determine the next mesh and thus the availability of these more reliable maximum defect estimates can lead to improved meshes. As well, when such estimates are available, the code can terminate with more confidence that the defect is bounded throughout the problem domain by the user-prescribed tolerance. In this paper we employ a boot-strapping approach to derive interpolants that allow asymptotically correct defect estimates. Numerical results are included to demonstrate the validity of this approach.
The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK... more The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODE's more than fteen years ago. During the last decade, there has been considerable investigation of the use of these schemes in the numerical solution of boundary value ODE problems, where their e cient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes. In fact, recent work in this area has seen the development of some software packages for boundary value ODE's based on these schemes. Unfortunately, these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval, while the collocation schemes provide a natural continuous solution approximation. The availability of a continuous solution is important not only to the user of the software but also within the code itself, for example, in estimation of errors, defect control, mesh selection, and the provision of initial solution estimates for new meshes. An approach for the construction of a continuous solution approximation based on the MIRK schemes is suggested by recent work in the area of continuous extensions for explicit Runge-Kutta schemes for initial value ODE's. In this paper, we describe our work in the investigation of continuous versions of the MIRK schemes: (i) we give some lower bounds relating the stage order to the minimal number of stages for general IRK schemes, (ii) we establish lower bounds on the number of stages needed to derive continuous MIRK schemes of orders 1 through 6, (iii) we provide characterizations of such schemes having a minimal number of stages for each of these orders.
Efficient Classes of Runge-Kutta Methods for Two-Point Boundary Value Problems. The standard appr... more Efficient Classes of Runge-Kutta Methods for Two-Point Boundary Value Problems. The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn x s equations in order to calculate the stages of the method, where n is the number of differential equations and s is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicitly, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.
The successful use of mono-implicit Runge-Kutta methods has been demonstrated by several research... more The successful use of mono-implicit Runge-Kutta methods has been demonstrated by several researchers who have employed these methods in software packages for the numerical solution of boundary value ordinary di erential equations. However, these methods are only applicable to rst order systems of equations while many boundary value systems involve higher order equations. While it is straightforward to convert such systems to rst order, several advantages, including substantial gains in e ciency, higher continuity of the approximate solution and lower storage requirements, are realized when the equations can be treated in their original higher order form. In this paper, we consider generalizations of mono-implicit Runge-Kutta methods, called monoimplicit Runge-Kutta-Nystrom methods, suitable for systems of second order ordinary di erential equations, having the general form, y 00 (t) = f(t; y(t); y 0 (t)), and derive optimal symmetric methods of orders two, four, and six. We also in...
The BACOL software package, which employs high order methods in time and space to adaptively cont... more The BACOL software package, which employs high order methods in time and space to adaptively control spatial and temporal errors in a method-of-lines approach, has been shown to be significantly more efficient than existing codes for the accurate numerical solution of systems of parabolic PDEs in one space dimension. In BACOL, the collocation spatial discretization gives a system of differential-algebraic equations (DAEs) which is treated using the DAE solver DASSL. Since DASSL employs backward differentiation formulas (BDFs), each spatial remeshing requires an interpolation of previous solution values. In addition, for DAE systems whose Jacobians have eigenvalues on the imaginary axis, such as those arising from Schrödinger problems, DASSL performs inefficiently since the higher order BDFs are not A-stable. In this paper, we describe a new software package, BACOLR, which addresses these issues by using RADAU5, a DAE solver based on an A-stable, (one-step) implicit RungeKutta method...
BACOL is a software package for the numerical solution of systems of one-dimensional parabolic pa... more BACOL is a software package for the numerical solution of systems of one-dimensional parabolic partial differential equations (PDEs) that has been shown to be superior to other similar packages, especially for problems exhibiting sharp spatial layer regions where a stringent tolerance is imposed. BACOL, based on a method-of-lines algorithm, features adaptive control of a high order estimate of the spatial error. (Adaptive control of the temporal error in the numerical solution of the system of differential-algebraic equations (DAEs), arising from a B-spline Gaussian collocation spatial discretization, is provided by the underlying DAE solver, DASSL.) The spatial error estimate for the collocation solution computed by the code is obtained by computing a second collocation solution, which involves a substantial cost the execution time and memory usage are almost doubled. In this report we discuss BACOLI, a new version of BACOL that computes only one collocation solution and uses effic...
BACOL and BACOLR are B-spline Gaussian collocation method-of-lines packages for the numerical sol... more BACOL and BACOLR are B-spline Gaussian collocation method-of-lines packages for the numerical solution of systems of one-dimensional parabolic partial differential equations (PDEs). In previous studies, they were shown to be superior to other similar packages, especially for problems exhibiting sharp spatial layer regions where a stringent tolerance is imposed. A significant feature of these solvers is that, in addition to the temporal error control provided by the underlying time-integrator, they adapt the spatial mesh to control a high order estimate of the spatial error. In addition to computing a primary collocation solution of a given spatial order, the BACOL/BACOLR codes also compute, at a substantial cost, a secondary collocation solution of one higher order, and then the difference between the two collocation solutions is used to give an estimate of the leading order term in the error for the lower order solution. In this paper we consider an approach in which the computatio...
Error control software packages based on Gaussian collocation have been widely used for the numer... more Error control software packages based on Gaussian collocation have been widely used for the numerical solution of boundary value ODEs (BVODEs) and 1D parabolic time-dependent PDEs (1D PDEs) for several decades. These robust and efficient packages are among the best available for these problem classes. In this paper, we survey error control Gaussian collocation software for BVODEs and 1D PDEs and provide an overview of recent work involving the development of two new packages, one for each problem class. The first is an updated version of the well-known COLSYS/COLNEW package for BVODEs. The second is the newest member of the BACOL family of software packages for 1D PDEs. We briefly review the underlying numerical algorithms employed in these packages and then provide numerical results that show the superiority of the new packages compared to previously released packages from each software class.
Computers & Mathematics with Applications, 2019
B-spline Gaussian collocation software has been widely used in the numerical solution of boundary... more B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and partial differential equations (PDEs) in one space dimension (1D) for many years. The software package, BACOL, developed over a decade ago, was one of the first 1D PDE packages to provide both temporal and spatial error control. A new package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages implies that extensive testing and analysis of the test results is an essential factor in their development. In this paper, we investigate the performance of the BACOL and BACOLI packages with respect to several important machine independent algorithmic measures and examine the effectiveness of the new error estimation and error control strategies. We also investigate the influence of the choice of the degree of the B-splines on the efficiency and reliability of the solvers. These results will provide new insights into how to improve BACOLI, lead to improvements in the Gaussian collocation BVODE solvers, COLSYS and COLNEW, and guide the further development of B-spline Gaussian collocation software with error control for 2D PDEs.
This paper considers the use of a superconvergent interpolant (SCI) for spatial error estimation ... more This paper considers the use of a superconvergent interpolant (SCI) for spatial error estimation when Gaussian collocation is employed as the spatial discretization scheme in a method-of-lines algorithm for the numerical solution of a system of one-dimensional parabolic partial differential equations (PDEs). Gaussian collocation is a popular approach for the spatial discretization of parabolic PDEs, and at certain points within the problem domain, the collocation solution is superconvergent. This paper describes how an interpolant based on these superconvergent values can be used to provide an efficient error estimate for the collocation solution. We implement this scheme within a modified version of the collocation PDE solver, BACOL. The original BACOL code obtains a spatial error estimate by computing a second global collocation solution of one higher order of accuracy. We show that the SCI based error estimation approach can provide spatial error estimates of comparable accuracy to those currently computed by BACOL, but at a much lower cost.
We consider systems of parabolic partial differential equations in one space variable, for which ... more We consider systems of parabolic partial differential equations in one space variable, for which we describe a method of lines algorithm based on monomial spline collocation for the discretization of the spatial domain. While the usual application of this technique transforms the system of partial differential equations into a system of time dependent ordinary differential equations which can be integrated by standard initial value solvers, our approach leads to coupled systems of differential-algebraic equations. These equations are solved using a well known package, DASSL, which we have modified to take advantage of the special structure of the Jacobians which arise.
Advances in Applied Mathematics and Mechanics, 2013
In this paper we describe newB-spline Gaussian collocation software for solving two-dimensional p... more In this paper we describe newB-spline Gaussian collocation software for solving two-dimensional parabolic partial differential equations (PDEs) defined over a rectangular region. The numerical solution is represented as a bi-variate piecewise polynomial (using a tensor productB-spline basis) with time-dependent unknown coefficients. These coefficients are determined by imposing collocation conditions: the numerical solution is required to satisfy the PDE and boundary conditions at images of the Gauss points mapped onto certain subregions of the spatial domain. This leads to a large system of time-dependent differential algebraic equations (DAEs) which is solved using the DAE solver, DASPK. We provide numerical results in which we use the new software, called BACOL2D, to solve three test problems.
A long-standing open question associated with the use of collocation methods for boundary value o... more A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obtained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h 2k), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C 0 continuous solution approximation that has a global error of O(h k+1). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C 1 continuous "superconvergent" interpolants whose global errors are O(h 2k). The key ideas are to use the theoretical framework of continuous Runge-Kutta schemes and to augment the collocation solution with inexpensive monoimplicit Runge-Kutta stages. Specific schemes are derived for k = 1, 2, 3, and 4. Numerical results are provided to support the theoretical analysis.
Proceedings of the 21st Koli Calling International Conference on Computing Education Research, 2021
Here we describe a project whose goal is to address issues associated with student takeaway, i.e.... more Here we describe a project whose goal is to address issues associated with student takeaway, i.e., enduring learning, in an introductory course in Numerical Analysis/Scientific Computing (NA/SC) commonly taught in undergraduate programs in Computer Science, Mathematics, and Engineering. We have employed the well-known framework of Threshold Concepts (TCs) in order to identify essential "takeaway" concepts in an introductory NA/SC course. We report on the four TCs we have proposed for NA/SC and discuss how the TC framework can be used to improve student takeaway.
Recent investigations of discretization schemes for the efficient numerical solution of boundary ... more Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C 1 continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. N...
. The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (I... more . The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODE's more than fifteen years ago. During the last decade, there has been considerable investigation of the use of these schemes in the numerical solution of boundary value ODE problems, where their efficient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes. In fact, recent work in this area has seen the development of some software packages for boundary value ODE's based on these schemes. Unfortunately, these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval, while the collocation schemes provide a natural continuous solution approximation. The availability of a continuous solution is important not only to the user of the software but also within the code itse...
A long-standing open question associated with the use of collocation methods for boundary value o... more A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, ob-tained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h 2k), where k is the number of collocation points used per subinterval and h is the subin-terval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C 0 continuous solution approximation that has a global error of O(h k+1). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C 1 continuous "superconvergent" interpolants whose global errors are O(h 2k). The key ideas are to use the theo-retical f...
A popular approach to the numerical solution of boundary value ODE problems involves the use of c... more A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge-Kutta formulas, known as mono-implicit Runge-Kutta (MIRK) formulas, which can be implemented at a lower cost per step than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather, only a discrete approximation at certain points within the problem interval is obtained. However, recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge-Kutta formula. These ideas have recently been extended to develop continuous extensions of the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on the continuous MIRK formulas, as an alternative to the standard use of global error control, as the basis for termination and mesh redistribution criteria.
The defect of a continuous approximate solution to an ODE is the amount by which that approximati... more The defect of a continuous approximate solution to an ODE is the amount by which that approximation fails to satisfy the ODE. A number of studies have explored the use of asymptotically correct defect estimates in the numerical solution of initial value ODEs (IVODEs). By employing an appropriately constructed interpolant to an approximate discrete solution to the ODE, various researchers have shown that it is possible to obtain estimates of the local error and/or the maximum defect that are asymptotically correct on each step, as the stepsize h → 0. In this paper, we investigate the usefulness of asymptotically correct defect estimates for defect control in boundary value ODE (BVODE) codes. In the BVODE context, for a sequence of meshes which partition the problem interval, one computes a discrete numerical solution, constructs an interplant, and estimates the maximum defect. The estimates (typically obtained by sampling the defect at a small number of points on each subinterval of the mesh) are used in a redistribution process to determine the next mesh and thus the availability of these more reliable maximum defect estimates can lead to improved meshes. As well, when such estimates are available, the code can terminate with more confidence that the defect is bounded throughout the problem domain by the user-prescribed tolerance. In this paper we employ a boot-strapping approach to derive interpolants that allow asymptotically correct defect estimates. Numerical results are included to demonstrate the validity of this approach.
The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK... more The mono-implicit Runge-Kutta (MIRK) schemes, a subset of the family of implicit Runge-Kutta (IRK) schemes, were originally proposed for the numerical solution of initial value ODE's more than fteen years ago. During the last decade, there has been considerable investigation of the use of these schemes in the numerical solution of boundary value ODE problems, where their e cient implementation suggests that they may provide a worthwhile alternative to the widely used collocation schemes. In fact, recent work in this area has seen the development of some software packages for boundary value ODE's based on these schemes. Unfortunately, these schemes lead to algorithms which provide only a discrete solution approximation at a set of mesh points over the problem interval, while the collocation schemes provide a natural continuous solution approximation. The availability of a continuous solution is important not only to the user of the software but also within the code itself, for example, in estimation of errors, defect control, mesh selection, and the provision of initial solution estimates for new meshes. An approach for the construction of a continuous solution approximation based on the MIRK schemes is suggested by recent work in the area of continuous extensions for explicit Runge-Kutta schemes for initial value ODE's. In this paper, we describe our work in the investigation of continuous versions of the MIRK schemes: (i) we give some lower bounds relating the stage order to the minimal number of stages for general IRK schemes, (ii) we establish lower bounds on the number of stages needed to derive continuous MIRK schemes of orders 1 through 6, (iii) we provide characterizations of such schemes having a minimal number of stages for each of these orders.
Efficient Classes of Runge-Kutta Methods for Two-Point Boundary Value Problems. The standard appr... more Efficient Classes of Runge-Kutta Methods for Two-Point Boundary Value Problems. The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn x s equations in order to calculate the stages of the method, where n is the number of differential equations and s is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicitly, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.
Uploads
Papers by Paul Muir