ISRAEL

Corrigé de la PC3 : Schémas numériques pour leséquations hyperboliques linéaires : le cas de l’équationdes ondes19 Mai 2011EXERCICE 1 (UN SCHÉMA POUR L’ÉQUATION DES ONDES.)On s’intéresse à l’approximation de l’équation des ondes 1D :∂ 2 u∂t − ∂2 u2 c2∂x = 0 x ∈ IR, t > 0, 2 (1)On rappelle que, pour toute fonction u suffisamment régulière :u(x + h) − 2u(x) + u(x − h)h 2On introduit l’opérateur aux différences finies := d2 u h2 d 4 u(x) +dx2 12 dx (x) + 4 O(h4 ) (2)A 1 2 u(x + h) − 2u(x) + u(x − h)hu(x) = − ch 2et on considère le schéma numérique, appelé schéma saute-moutonu n+1h− 2u n h + un−1 h∆t 2 + A 1 hu n h = 0 (3)Question 1. Expliquer sans calculs pourquoi ce schéma est consistant. Étudier lavitesse de propagation numérique et en déduire une condition nécessaire de convergencedu schéma.Corrigé de la question 1 : Il suffit d’utiliser l’expression (2) pour déduire que leschéma est consistant, d’ordre 2 en temps et en espace.De plus, le support de la solution numérique se propage à vitesse finie au sens oùsupp u n h ⊂ [a, b] =⇒ supp u n+1h⊂ [a − h, b + h].Autrement dit, la vitesse de propagation numérique est donnée par :V num = h ∆tLa vitesse de propagation des ondes étant c ici, une condition nécessaire de stabilitéest V num ≥ c, ce qui se réécrit :c∆th ≤ 11

CONTENTSAvi Soffer Designs .............................................................................................................4Bat-Ami.....................................................................................................................5Beit-Nir Jewelry......................................................................................................................6Jolie Diamond &Jewelry ..................................................................................................7K.G.H. Jewelry Ltd. .............................................................................................................8Leon Israel Designs ............................................................................................................9M.I. Jewelry by Marika......................................................................................................10Majestic.........................................................................................................................11R.A.N . GOV Jewelry ........................................................................................................12Rashbel................................................................................................................................13Roman Glass / Sapirim Rahav.............................................................................14Sexy Silver / Orit Schatzman ....................................................................................15Shablool Silver Jewelry Design Ltd. .....................................................................16Shen-hen Henia ................................................................................................................17Tamir Zuman Jewelry .....................................................................................................18Contents | PDF |3

Booth S10300Avi Soffer is a driving force in the jewelry industry,constantly innovating to complement changing trends. For35 years, Avi has been creating unique, eye-catching worksof wearable art, implementing semi-precious gemstonesand pearls set in sterling silver and gold.Contact DetailsMr. Avi Soffer, President+972 3 752 7111 ++972 3 752 7115avi@soffer.co.il www.avisoffer.comContents | PDF |4

Booth S10209Bat-Ami offers contemporary electroformed sterling silverjewelry, designed and hand-crafted at our own factory inTel Aviv: sophisticated, lightweight pure silver “wearable art”complemented by leather cords, semi-precious stones orrecycled glass.Highly polished or brushed, we do everything with silver.Contact DetailsMrs. Einat Kartin, Marketing Director+972 3 561 6004 ++972 3 562 1306bat-ami@bat-ami.com www.bat-ami.comContents | PDF |5

Booth B6070Beit-Nir Jewelry was founded in 1964 by a small group ofartisans at Kibbutz Beit-Nir in Israel’s Negev Desert.Our jewelry collection is imbued with textures and shapesinspired by the beautiful environment that surrounds us, andby the social atmosphere of our kibbutz community.Contact DetailsMr. Shimon Samama, Manager+972 8 687 4313 ++972 8 681 2249info@beitnir-jewellery.com www.beitnir-jewelry.comContents | PDF |6

Booth B10479Jolie Diamond & Jewelry offers a rich collection of uniquenew designs. We specialize in pave and round invisiblesettings using diamonds and semi-precious stones. We alsomanufacture custom and private designs.Contact DetailsMr. Elias K. Haddad, Owner+972 4 655 2490 ++972 4 600 1787info@joliediamond.com www.joliediamond.comContents | PDF |7

Booth B2128KGH Jewellery manufactures pearl gold jewellery andfindings specializing in clasps. Our collection, renowned forquality and easy to use reliable mechanisms, includes a widerange of clasps, earrings, pendants and other pearl jewellerywith or without diamonds and in different finishes.Contact DetailsMr. Eitan Konig, Marketing Manager+972 3 752 5019kgh@kgh.co.il www.kgh.co.ilContents | PDF |8

Booth S10205Leon Israel Designs.The jewelry in this collection, designed by Leon Israel, ischaracterized by soft flowing lines and deep textures, inwhich colors and forms of gold, precious stones and pearlswork harmoniously together, creating jewelry to meet themost discriminating taste.Contact DetailsMr. Leon Israel, Mrs. Miriam Israel, Owners+972 3 682 5443 ++972 3 682 5443Lidesign@netvision.net.ilContents | PDF |9

Booth S107 07M.I. Jewelry by MarikaMarika designs and produces handmade 14 and 18 kt. goldjewelry set with diamonds and natural stones. Chief designerMarika Lev takes inspirations from nature and applies them tonew trends. The Desert Gold line is colored and shaped to awoman’s body, giving it a unique look.Contact DetailsMr. Guy Hertz, Marketing Manager+972 9 885 4007, +972 54 253 9191 ++972 9 885 4010Guy@mijbymarika.com www.mijbymarika.comContents | PDF | 10

Booth B 3659Majestic manufactures high quality diamond jewelry soughtfor its unique, patented designs that set stones invisibly to aprecision of 1/100 mm. Each Majestic jewel is created froma number of small round diamonds that sparkle with thesplendor of a single large diamond.Contact DetailsMr. Ron Stschik+972 3 612 6472 / 3 +972 3 612 6471majestic@netvision.co.il www.majestic.co.ilContents | PDF | 11

Booth LUX433AR.A.N. GOV Jewelry, founded 50 years ago, designs andmanufactures high quality gold jewelry, set with diamonds,precious- and semi-precious stones and pearls.Contact DetailsMrs. Nitza Drori, Marketing Manager+972 3 518 6010 ++972 3 683 5030nitza@rangov.comContents | PDF | 12

Booth B6072Rashbel produces hand-made chains in GF & S925, as wellas pendants, charms, rings & special bezel cups, uniquegallery strips, fancy pattern wire, pearl & bead wire in gold,GF, S925, brass and more.We have been exporting worldwide for over 30 years.Contact DetailsMr. Shimon Lavi, CEO+972 3 620 1920 ++972 3 620 4899order@rashbel.biz www.rashbel.bzContents | PDF | 13

Booth B6066Roman Glass / Sapirim Rahav specializes in setting ancientglass in silver jewelry. Every piece of glass is unique and eachitem comes with a certificate of authenticity. We also offersterling silver jewelry set with stones.Contact DetailsMs. Roni Keren, Marketing Executive+972 8 681 6611 ++972 8 681 6622rg@romanglass.co.il www.romanglass.comContents | PDF | 14

Booth B6068Sexy Silver / Orit Schatzman was established in 2001 byOrit and Zvi Schatzman. Our jewelry is designed by Orit, andis made of 925 sterling with 24k gold vermeil available.Customers delight in our unique stone setting with semipreciousstones and Swarovski crystals.Contact DetailsMs. Orit Shatzman, President+972 9 741 6391 ++972 9 760 3031zmbs@012.net.il http://oritschatzmanstore.comContents | PDF | 15

Booth S10201Shablool Silver Jewelry Design - The company designsand manufactures 925 sterling silver jewelry combined withsemi-precious stones and 9k gold, producing rings, earrings,bracelets, necklaces, watches and pins. We supply anyquantity at top qualityContact DetailsMr. David Moss, Marketing Manager+972 4 652 1415 ++972 4 652 7387david@shablool.com www.shablool.comContents | PDF | 16

Booth S10307Shen-hen Henia creates transformable jewelry that differsespecially in the ideas embodied within it and in the way it isworn. Our silver and gold jewelry with diamond and preciousstones is of the highest quality.Large production runs of all designs available.Contact DetailsMs. Henia Shnitzer, President+972 4 953 2531 ++972 4 953 1835heniashn2002@gmail.com www.shen-hen.comContents | PDF | 17

Booth S10207Tamir Zuman produces - from design through casting tofinish - an extensive range of jewelry items and sterlingsilver watches. Our collections are distinctive and fresh, andare made with the finest materials and craftsmanship.We are known for excellent customer service.Contact DetailsMr. Tamir Zuman, President+972 8 940 8268 ++972 8 940 8344tamir@tamirzuman.com www.tamirzuman.comContents | PDF | 18

For further Informationplease contact:Mr. Roy VigodskyBusiness Development Manager,Jewelry and Giftware+972 3 514 2862royv@export.gov.ilwww.export.gov.ilContents | PDF |