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Special Section on Sparse Matrix Algorithms on Supercomputers 超级计算机上的稀疏矩阵算法专题
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-11-01 DOI: 10.1137/0910068
C. Gear, H. Simon
{"title":"Special Section on Sparse Matrix Algorithms on Supercomputers","authors":"C. Gear, H. Simon","doi":"10.1137/0910068","DOIUrl":"https://doi.org/10.1137/0910068","url":null,"abstract":"","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125903608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mathematical structure of compositional reservoir simulation 组份油藏模拟的数学结构
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910049
J. Trangenstein, J. Bell
{"title":"Mathematical structure of compositional reservoir simulation","authors":"J. Trangenstein, J. Bell","doi":"10.1137/0910049","DOIUrl":"https://doi.org/10.1137/0910049","url":null,"abstract":"In this paper multicomponent two-phase isothermal fluid flow in petroleum reservoirs is described. The fluid-flow model consists of component conservation equations, Darcy's law for the volumetric flow rates, balance between the fluid volume and the rock void, and the conditions of thermodynamic equilibrium that determine the distribution of the chemical components into phases. Thermodynamic equilibrium is described by means of a mathematical model for the chemical potentials of each component in each phase of the fluid. The flow equations are manipulated to form a pressure equation and a modified component-conservation equation; these form the basis for the sequential method. It is shown that the pressure equation is parabolic under reasonable assumptions on the thermodynamic equilibrium model, and that the component-conservation equations are hyperbolic in the absence of diffusive forces such as capillary pressure and mixing. A numerical method based on the sequential formulation of the flow equations is outlined and used to illustrate the kinds of flow behavior that occur during miscible gas injection.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"115 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129047317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 76
Asymptotic expansions of the global discretization error for stiff problems 刚性问题全局离散化误差的渐近展开式
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910055
W. Auzinger, R. Frank
{"title":"Asymptotic expansions of the global discretization error for stiff problems","authors":"W. Auzinger, R. Frank","doi":"10.1137/0910055","DOIUrl":"https://doi.org/10.1137/0910055","url":null,"abstract":"The existence of asymptotic expansions of the global discretization error for a general class of nonlinear stiff differential equations [ y'(t) = A(t)y(t) + varphi (t,y(t)), ] where $A(t)$ has a “stiff spectrum” characterized by a small parameter $varepsilon $ and where $varphi (t,y)$ is smooth, is discussed. The following methods are considered: implicit Euler, implicit midpoint, and trapezoidal rules. In strongly stiff situations ($varepsilon $ significantly smaller than the stepsize h) the implicit Euler scheme admits a full asymptotic expansion; the same is true for the midpoint rule and for the trapezoidal rule under certain coupling conditions. In those strongly stiff cases where a full expansion does not exist for the midpoint or trapezoidal rule, the remainder term is of a reduced order but shows a regular, oscillating behavior that is described in detail. In mildly stiff situations, order reductions of the remainder term inevitably occur in any case after the start or after the change of stepsize but—as can be shown by discrete singular perturbation techniques—these order reductions are rapidly damped out as the integration proceeds. Results are illustrated by various numerical examples; in particular, numerical experience with extrapolation and defect correction is reported.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124333119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
To overlap or not to overlap: a note on a domain decomposition method for elliptic problems 重叠或不重叠:关于椭圆问题的一个区域分解方法的注解
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910063
P. Bjørstad, O. Widlund
{"title":"To overlap or not to overlap: a note on a domain decomposition method for elliptic problems","authors":"P. Bjørstad, O. Widlund","doi":"10.1137/0910063","DOIUrl":"https://doi.org/10.1137/0910063","url":null,"abstract":"More than one hundred years ago, H.A. Schwarz introduced a domain decomposition method in which the original elliptic equation is solved on overlapping subregions, one after another, in an iterative process. A few years ago, Chan and Resasco, introduced a method that they classified as a domain decomposition method using nonoverlapping subdomains. In this note, it is shown that their method is an accelerated version of the classical method. It is also shown that the error propagation operator of the method can be expressed in terms of Schur complements of certain stiffness matrices and that techniques previously developed for the study of iterative substructuring algorithms can be used to derive estimates on the rate of convergence.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117140356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 73
Defect estimation in Adams PECE codes Adams PECE代码的缺陷估计
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910056
D. Higham
{"title":"Defect estimation in Adams PECE codes","authors":"D. Higham","doi":"10.1137/0910056","DOIUrl":"https://doi.org/10.1137/0910056","url":null,"abstract":"Many modern codes for solving the nonstiff initial value problem $y'(x) - f(x,y(x)) = 0,y(a)$ given, $a leqq x leqq b$, produce, in addition to a discretised solution, a function $p(x)$ that approximates $y(x)$ over $[a,b]$. The associated defect $delta (x): = p'(x) - f(x,p(x))$ is a natural measure of the error. In this paper the problem of reliably estimating the defect in Adams PECE methods is considered. Attention is focused on the widely used Shampine–Gordon variable order, variable step code fitted with a continuously differentiable interpolant $p(x)$ due to Watts and Shampine [SIAM .J. Sci. Statist. Comput, 7 (1986), pp. 334–345]. It is shown that over each step an asymptotically correct estimate of the defect can be obtained by sampling at a single, suitably chosen point. It is also shown that a valid “free” estimate can be formed without recourse to sampling. Numerical results are given to support the theory.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121815297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Stability properties of backward euler multirate formulas 后向欧拉多速率公式的稳定性
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910059
S. Skelboe
{"title":"Stability properties of backward euler multirate formulas","authors":"S. Skelboe","doi":"10.1137/0910059","DOIUrl":"https://doi.org/10.1137/0910059","url":null,"abstract":"Stability properties of multirate formulas cannot be analyzed by a scalar test equation but require at least one equation for each different steplength. This paper generalizes the concept of absolute stability and A-stability for backward Euler multirate formulas. Stability theorems for multirate methods with two and three different steplengths are given, while a general result for an arbitrary number of different steplengths is the topic of future research.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131363763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 36
The Effect of Changing the Stepsize in Linear Multistep Codes 改变步长对线性多步码的影响
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910060
L. Shampine, P. Bogacki
{"title":"The Effect of Changing the Stepsize in Linear Multistep Codes","authors":"L. Shampine, P. Bogacki","doi":"10.1137/0910060","DOIUrl":"https://doi.org/10.1137/0910060","url":null,"abstract":"In the usual convergence theory for linear multistep methods, a constant stepsize h is used. For a method of order p, the discretization error is proportional to $h^{p + 1} $. In a variable step code, it is necessary to predict what the discretization error would be if the stepsize were changed to $rh$. It is usual to say that the observed error will be altered by a factor of $r^{p + 1} $. Unfortunately this is not correct for multistep methods. The discrepancy arises in the fact that the usual theory does not model the way variable stepsize codes actually work. In this paper the correct behavior is determined for important classes of formulas and ways of changing stepsize.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"33 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120855904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 12
A new class of parallel algorithms for solving systems of linear equations 求解线性方程组的一类新的并行算法
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910051
K. Jainandunsing, E. Deprettere
{"title":"A new class of parallel algorithms for solving systems of linear equations","authors":"K. Jainandunsing, E. Deprettere","doi":"10.1137/0910051","DOIUrl":"https://doi.org/10.1137/0910051","url":null,"abstract":"In this paper a class of novel feed-forward direct methods is presented for solving nonsingular systems of linear equations. The computational complexity of these methods is in the order of an $LU$, $QR$, or $LL^t $ matrix factorization. This is also true for the complexity of their systolic implementations. Unlike the direct methods of factorization followed by backsubstitution, the systolic implementations of the novel methods do not suffer from the backsubstitution bottleneck. A numerically stable and robust method, which uses only Givens rotations as elementary operations, is included in the class.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126104227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 42
VODE: a variable-coefficient ODE solver VODE:一个变系数ODE求解器
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910062
P. Brown, G. D. Byrne, A. Hindmarsh
{"title":"VODE: a variable-coefficient ODE solver","authors":"P. Brown, G. D. Byrne, A. Hindmarsh","doi":"10.1137/0910062","DOIUrl":"https://doi.org/10.1137/0910062","url":null,"abstract":"VODE is a new initial value ODE solver for stiff and nonstiff systems. It uses variable-coefficient Adams-Moulton and Backward Differentiation Formula (BDF) methods in Nordsieck form, as taken from the older solvers EPISODE and EPISODEB, treating the Jacobian as full or banded. Unlike the older codes, VODE has a highly flexible user interface that is nearly identical to that of the ODEPACK solver LSODE.In the process, several algorithmic improvements have been made in VODE, aside from the new user interface. First, a change in stepsize and/or order that is decided upon at the end of one successful step is not implemented until the start of the next step, so that interpolations performed between steps use the more correct data. Second, a new algorithm for setting the initial stepsize has been included, which iterates briefly to estimate the required second derivative vector. Efficiency is often greatly enhanced by an added algorithm for saving and reusing the Jacobian matrix J, as it occurs in the Newton m...","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131308474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1586
On symmetric schemes and differential-algebraic equations 关于对称格式与微分代数方程
Siam Journal on Scientific and Statistical Computing Pub Date : 1989-09-01 DOI: 10.1137/0910054
U. Ascher
{"title":"On symmetric schemes and differential-algebraic equations","authors":"U. Ascher","doi":"10.1137/0910054","DOIUrl":"https://doi.org/10.1137/0910054","url":null,"abstract":"An example is given that demonstrates a potential risk in using symmetric difference schemes for initial value differential-algebraic equations (DAEs) or for very stiff ordinary differential equations (ODEs). The basic difficulty is that the stability of the scheme is controlled by the stability of an auxiliary (ghost) ODE problem that is not necessarily stable even when the given problem is.The stability of symmetric schemes is better understood in the context of boundary value problems. In this context, such schemes are more naturally applied as well. For initial value problems, better alternatives may exist. A computational algorithm is proposed for boundary value index-1 DAEs.","PeriodicalId":200176,"journal":{"name":"Siam Journal on Scientific and Statistical Computing","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1989-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127322430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 45
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