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Introduction to Partial Differential Equations: A Computational Approach

Aslak Tveito Ragnar Winther

Resumen/Descripción – provisto por la editorial

No disponible.

Palabras clave – provistas por la editorial

Analysis; Partial Differential Equations; Computational Science and Engineering

Disponibilidad
Institución detectada Año de publicación Navegá Descargá Solicitá
No detectada 2005 SpringerLink

Información

Tipo de recurso:

libros

ISBN impreso

978-3-540-22551-5

ISBN electrónico

978-3-540-26740-9

Editor responsable

Springer Nature

País de edición

Reino Unido

Fecha de publicación

Información sobre derechos de publicación

© Springer-Verlag Berlin Heidelberg 2005

Cobertura temática

Tabla de contenidos

Setting the Scene

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 1-37

Two-Point Boundary Value Problems

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 39-86

The Heat Equation

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 87-116

Finite Difference Schemes for the Heat Equation

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 117-158

The Wave Equation

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 159-173

Maximum Principles

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 175-207

Poisson's Equation in Two Space Dimensions

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 209-244

Orthogonality and General Fourier Series

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 245-284

Convergence of Fourier Series

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 285-312

The Heat Equation Revisited

Aslak Tveito; Ragnar Winther

A safety-critical real-time computer system must provide its services with a dependability that is much better than the dependability of any one of its constituent components. This challenging goal can only be achieved by the provision of fault tolerance. The design of any fault-tolerant system proceeds in four distinct phases. In the first phase the fault hypothesis is shaped, i.e. assumptions are made about the types and numbers of faults that must be tolerated by the planned system. In the second phase an architecture is designed that tolerates the specified faults. In the third phase the architecture is implemented and the functions and fault-tolerance mechanisms are validated. Finally, in the fourth phase it has to be confirmed experimentally that the assumptions contained in the fault-hypothesis are met by reality. The first part of this contribution focuses on the establishment of a comprehensive fault hypothesis for safety-critical real-time computer systems. The size of the fault containment regions, the failure mode of the fault containment regions, the assumed frequency of the faults and the assumptions about error detection latency and error containment are discussed under the premise that in future a distributed system node is expected to be a system-on-a-chip (SOC). The second part of this contribution focuses on the implications that such a fault hypothesis will have on the future architecture of distributed safety-critical real-time computer systems in the automotive domain.

Pp. 313-336