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732
On Nonreflecting Boundary Conditions
 J. COMPUT. PHYS
, 1995
"... Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated ..."
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Cited by 219 (4 self)
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Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated condition. Second, the exact DtN boundary condition is derived for elliptic and spheroidal coordinates. Third, approximate local boundary conditions are derived for these coordinates. Fourth, the truncated DtN condition in elliptic and spheroidal coordinates is modified to remove difficulties. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions. Numerical results are presented to demonstrate the usefulness of these improvements.
ADIFOR  Generating Derivative Codes from Fortran Programs
, 1992
"... The numerical methods employed in the solution of many scientific computing problems require the computation of derivatives of a function f: Rn! Rm. Both the accuracy and the computational requirements of the derivative computation are usually of critical importance for the robustness and speed of ..."
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Cited by 190 (66 self)
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The numerical methods employed in the solution of many scientific computing problems require the computation of derivatives of a function f: Rn! Rm. Both the accuracy and the computational requirements of the derivative computation are usually of critical importance for the robustness and speed of the numerical solution. ADIFOR (Automatic Differentiation In FORtran) is a source transformation tool that accepts Fortran 77 code for the computation of a function and writes portable Fortran 77 code for the computation of the derivatives. In contrast to previous approaches, ADIFOR views automatic differentiation as a source transformation problem. ADIFOR employs the data analysis capabilities of the ParaScope Parallel Programming Environment, which enable us to handle arbitrary Fortran 77 codes and to exploit the computational context in the computation of derivatives. Experimental results show that ADIFOR can handle reallife codes and that ADIFORgenerated codes are competitive with divideddifference approximations of derivatives. In addition, studies suggest that the source transformation approach to automatic differentation may improve the time to compute derivatives by orders of magnitude.
ADOLC: A Package for the Automatic Differentiation of Algorithms Written in C/C++
, 1995
"... The C++ package ADOLC described here facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The resulting derivative evaluation routines may be called from C/C++, Fortran, or any other language that can be linked ..."
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Cited by 190 (26 self)
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The C++ package ADOLC described here facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The resulting derivative evaluation routines may be called from C/C++, Fortran, or any other language that can be linked with C. The numerical values of derivative vectors are obtained free of truncation errors at a small multiple of the run time and randomly accessed memory of the given function evaluation program. Derivative matrices are obtained by columns or rows. For solution curves defined by ordinary differential equations, special routines are provided that evaluate the Taylor coefficient vectors and their Jacobians with respect to the current state vector. The derivative calculations involve a possibly substantial (but always predictable) amount of data that are accessed strictly sequentially and are therefore automatically paged out to external files.
ImplicitExplicit Methods For TimeDependent PDEs
 SIAM J. NUMER. ANAL
, 1997
"... Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection ..."
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Cited by 178 (6 self)
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Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reactiondiffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods. For the prototype linear advectiondiffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay of high freque...
Exponential integrators for large systems of differential equations,
 SIAM J. Sci. Comput.
, 1998
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Fourthorder time stepping for stiff PDEs
 SIAM J. SCI. COMPUT
, 2005
"... A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made ..."
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Cited by 94 (3 self)
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A modification of the exponential timedifferencing fourthorder Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential timedifferencing (ETD) scheme against the competing methods of implicitexplicit differencing, integrating factors, timesplitting, and Fornberg and Driscoll’s “sliders ” for the KdV, Kuramoto–Sivashinsky, Burgers, and Allen–Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best.
ImplicitExplicit RungeKutta schemes and applications to hyperbolic systems with relaxation
 Journal of Scientific Computing
, 2000
"... We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The sch ..."
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Cited by 83 (14 self)
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We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by finite difference discretization with Weighted Essentially Non Oscillatory (WENO) reconstruction. After a brief description of the mathematical properties of the schemes, several applications will be presented. Keywords: RungeKutta methods, hyperbolic systems with relaxation, stiff systems, high order shock capturing schemes. AMS Subject Classification: 65C20, 82D25 1
Adjoint sensitivity analysis for differentialalgebraic equations: algorithms and software
, 2002
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Exponential integrators
, 2010
"... In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential eq ..."
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Cited by 68 (5 self)
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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in
Notes on triangular sets and triangulationdecomposition algorithms II: Differential Systems
 SYMBOLIC AND NUMERICAL SCIENTIFIC COMPUTING
, 2003
"... This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We em ..."
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Cited by 65 (8 self)
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This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.