Numerical Methods for Particle Tracing in Vector Fields
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1 On-Line Visualization Notes Numerical Methods for Particle Tracing in Vector Fields Kenneth I. Joy Visualization and Grahics Research Laboratory Deartment of Comuter Science University of California, Davis Overview If we consider a vector field in sace and we have a rectilinear volume whose vertices are samled from this vector field, one method of visualizing the field is to lace articles in the field and animate their motion through the field. This time animation is the basis for a number of visualization methods streamlines, stream surfaces, and stream ribbons. In this aer we review the numerical techniques necessary for tracing the ath of a article through the field. We will study these techniques in two-dimensional vector fields, but they are directly extensible to three-dimensional fields. Interolating the Vectors Consider a oint inside one cell of our rectilinear volume. This cell has four corner oints 0,0, 1,0, 0,1, and 1,1, each of which has an associated vector. v 0,0, v 1,0, v 0,1, and v 1,1, resectively. We will assume that our cells have a unit width and height. 1 1 This is really no restriction, as we can scale our rectilinear volume such that the cells have this roerty.
2 Numerical Methods for Vector Fields Page 2 of 7 v 0,1 0,1 1,1 v 1,1 v v 0,0 v 0,0 1,0 u 1 v 1,0 We can calculate the vector at by using the values u and v and bilinearly interolating the vectors at the corners. Here v = (1 v) v 0 + v v 1 where v 0 = (1 u) v 0,0 + u v 1,0 v 1 = (1 u) v 0,1 + u v 1,1 or v = (1 v)(1 u) v 0,0 + (1 v)u v 1,0 + v(1 u) v 0,1 + vu v 1,1 Euler s Method Given a oint in a cell of our rectilinear grid, an obvious way to trace a article through a cell is to ste along the vector v associated with using a very short ste size t. The new oint 1 can be written as 1 = + t v We now calculate a oint 2 by steing along the vector associated with 1 (calculated by the bilinear interolation algorithm of the section above), and continue in this fashion. This is illustrated in the following figure.
3 Numerical Methods for Vector Fields Page 3 of 7 v This figure suggests the following strategy: (1) Select a ste size t; (2) Starting with the oint i, where 0 =, we erform the following iteration: Calculate v i by bilinearly interolating the vectors at the corners of the cell (Note: You first have to determine the cell in which i lies.), and Calculate i+1 = i + t v i This algorithm is called Euler s Algorithm. 2 It has an error estimate which is bounded by O( t 2 ) 3, which is not very good for the estimates that we require in visualization. The algorithm is mostly used as an initial aroximation method for the better algorithms discussed below. The Imroved Euler s Method The imroved Euler method utilizes Euler s method to redict the location of the new oint, but then utilizes more information from this oint to correct the initial guess. This method is illustrated in the following figure. v P i+1 v i P i+1 i+1 i 2 Leonhard Euler was a Swiss mathematician of the 18th century. 3 This estimate comes from Taylor s theorem which can be found in any elementary numerical analysis book.
4 Numerical Methods for Vector Fields Page 4 of 7 In this figure, we begin at the oint i. We use Euler s method to redict the location of the next oint in the iteration P i+1. We then use the calculated vector vp i+1 at P i+1 as additional information that can be used to better redict i+1. To obtain i+1, we begin at i and move half the ste size ( 1 2 t) along v i, then half the ste size along the vector v i+1 P. Mathematically, this can be written as i+1 = i t v i t vp i+1 = i t ( v i + v i+1 P ) This numerical method is called a redictor-corrector method (for obvious reasons). It has an error estimate which is bounded by O( t 3 ), which is substantially better than Euler s algorithm. The following illustration shows the difference between Euler s method and the imroved Euler method on our examle cell. The white dots are the aroximation due to Euler s method. The black dots are the aroximation due to the imroved Euler method. This imroved version is substantially used in the visualization community, but a better version, generated by two German mathematicians, Runge and Kutta, is most frequently used. The Runge-Kutta Algorithm The fourth-order Runge-Kutta method is similar to the imroved Euler method, but utilizes several redictor and corrector stes. The redictor values are illustrated in this following figure (We note that the icture was drawn with t = 1).
5 Numerical Methods for Vector Fields Page 5 of 7 v 1 i+1 v i v 2 i+1 v 3 i+1 1 i+1 2 i+1 3 i+1 i In this case, three vector redictors are calculated: v i+1 1 the vector corresonding to the oint i t v i; v i+1 2 the vector corresonding to the oint i t v1 i+1 ; and v i+1 3 the vector corresonding to the oint i + t v i+1 2. We note that each redictor is used to obtain the subsequent redictor values. These vectors are blended into the final result as follows: i+1 = i t v i t v1 i t v2 i t v3 i+1 which is shown in the following figure. v i v 1 i+1 v 2 i+1 v 3 i+1 i t 1 3 t 1 6 t i
6 Numerical Methods for Vector Fields Page 6 of 7 The formula can be simlified to i+1 = i t ( v i + 2 v i v2 i+1 + ) v3 i+1 which is the way it is usually written. This method seems comlex, but can be easily imlemented. The stes are the following: calculate the vector k 1 = t v i ; determine the vector v 1 i+1, which is the vector corresonding to the oint i k 1; calculate the vector k 2 = t v 1 i+1 ; determine the vector v 2 i+1, which is the vector corresonding to the oint i k 2; calculate the vector k 3 = t v 2 i+1 ; determine the vector v 3 i+1, which is the vector corresonding to the oint i + k 3 ; calculate the vector k 4 = t v i+1 3 ; and then calculate i+1 = i (k 1 + 2k 2 + 2k 3 + k 4 ) The differences between Euler s method, the imroved Euler method, and the Runge-Kutta method for our samle cell are shown in the following illustration. The white dots are the aroximations due to Euler s method, the gray dots are the aroximations due to the imroved Euler method, and the black dots are the aroximations due to the fourth-order Runge-Kutta method.
7 Numerical Methods for Vector Fields Page 7 of 7 The Runge-Kutta method resented here has an error bounded by O( t 4 ), which is significantly better than the other two algorithms. Summary We have resented three numerical algorithms that can be used for article tracing in vector fields. These algorithms, the Euler method, the imroved Euler method and the Runge-Kutta method, have errors bounded by O( t 2 ), O( t 3 ), and O( t 4 ), resectively. They are each easy to imlement, and the user can ick the algorithm necessary to achieve the errors desired by the alication. These algorithms lift easily to three-dimensional scalar fields. The only difference is that we must utilize trilinear interolation to calculate the vectors from the eight corner oints of a cell.
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