Simulation of phenomena characterized by partial differential equations
AssigneeThe Board of Trustees of the Leland Stanford Junior University
InventorsWonseok Shin, Shanhui Fan
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Abstract
Various aspects of the present disclosure are directed toward apparatus, methods, and articles of manufacture for simulating a physical phenomenon characterized by a set of partial differential equations that are reduced based on harmonic time dependence. The system of linear equations represent an operation in the differential equations, and are characterized by a coefficient matrix multiplied by an unknown column vector, with the product thereof being equal to a column vector. The system of linear equations is solved using at least one of three sets of operations.
Claims24
An apparatus for simulating a physical phenomenon characterized by a set of partial differential equations that are reduced based on harmonic time dependence, the apparatus comprising:
- a data computing circuit having
- stored therefor a system of linear equations representing an operation in the differential equations and in which a coefficient matrix is multiplied by an unknown column vector and with the product thereof being equal to a column vector, and
- stored therein a program for executing an algorithm which, when executed, solves the system of linear equations, wherein the algorithm sets forth at least one of the following three sets of operations:
- (i) multiplying the coefficient matrix by left and right scale-factor preconditioning matrices, and multiplying the column vector by the left scale-factor preconditioning matrix, each matrix including at least one of:
- entries of scale factors and combinations of scale factors of a perfectly matched layer,
- (ii) symmetrizing the coefficient matrix by multiplying the coefficient matrix by left and right diagonally-dominant preconditioning matrices while maintaining a condition number of the coefficient matrix within an order of magnitude of a condition number of the matrix prior to symmetrizing, and
- (iii) confining a majority of eigenvalues of the coefficient matrix in a half-plane having a boundary that crosses an origin of a complex plane.
A method for simulating a physical phenomenon characterized by a set of partial differential equations that are reduced based on harmonic time dependence, the method comprising:
- in a data computing circuit having stored therefor a system of linear equations representing an operation in the differential equations and in which a coefficient matrix is multiplied by an unknown column vector and with the product thereof being equal to a column vector, solving the system of linear equations by executing an algorithm that sets forth at least one of the following three sets of operations:
- multiplying the coefficient matrix by left and right scale-factor preconditioning matrices, and multiplying the column vector b by the left scale-factor preconditioning matrix, each matrix including at least one of:
- entries of scale factors and combinations of scale factors of a perfectly matched layer,
- symmetrizing the coefficient matrix by multiplying the coefficient matrix by left and right diagonally-dominant preconditioning matrices while maintaining a condition number of the coefficient matrix within an order of magnitude of a condition number of the matrix prior to symmetrizing, and
- confining a majority of eigenvalues of the coefficient matrix in a half-plane having a boundary that crosses an origin of a complex plane.
An article of manufacture for simulating a physical phenomenon characterized by a set of partial differential equations that are reduced based on harmonic time dependence, comprising:
- for execution by a processor having stored therefor a system of linear equations representing an operation in the differential equations and in which a coefficient matrix is multiplied by an unknown column vector and with the product thereof being equal to a column vector, a non-transitory processor-readable storage medium configured and arranged with program data that, when executed by the processor, causes the processor to solve the system of linear equations by at least one of:
- multiplying the coefficient matrix by left and right scale-factor preconditioning matrices, and multiplying the column vector b by the left scale-factor preconditioning matrix, each matrix including entries of scale factors or combinations of scale factors of a perfectly matched layer,
- symmetrizing the coefficient matrix by multiplying the coefficient matrix by left and right diagonally-dominant preconditioning matrices while maintaining a condition number of the coefficient matrix within an order of magnitude of a condition number of the matrix prior to symmetrizing, and
- confining a majority of eigenvalues of the coefficient matrix in a half-plane having a boundary that crosses an origin of a complex plane.
Dependencies are inferred from the claim text.